http://kumarmaths.weebly.com/ Core Mathematics Differentiation C differentiation Page
Differentiation C Specifications. By the end of this unit you should be able to : Use chain rule to find the derivative of composite functions. Find the derivative of products and quotients. Find the derivatives of trigonometric, logarithmic and exponential functions. Work will also include turning points and the equations of tangents and normals. C differentiation Page
The following differentials must be learnt. Function lnx e mx Sin t Cos t Tan t Sec t Cot t Cosec t Sin n t Cos n t Sin (g(t)) Cos (g(t)) Tan (g(t)) Differential x me mx Cos t -Sin t Sec t Sec t Tan t -Cosec t -Cosec t cot t nsin n- t Cos t -ncos n- t Sin t g (t)cos (g(t)) -g (t)sin (g(t)) g (t)sec (g(t)) Chain Rule If y is a function of v and v, in turn, is a function of x, then: dv dv Product Rule d uv dv u v du Quotient Rule d u v v du v dv u C differentiation Page
Composite Functions y = (mx + c) n nm(mx n c) y = (f(x)) n nf'(x) n (f(x)) The following example covers most of the ideas introduced above. Example Differentiate the following with respect to x simplify your answer as far as possible. a) (x + ln x) 7 b) 6 sin x + sec x c) x tan x d) x 5 e 5x + 5 e) 5 cos5x x C differentiation Page
d 7 a) (x + ln x) This is a composite function and chain rule must be used. By chain rule If y = (x + ln x) 7 let u = (x + ln x) So y = u 7 du not x x 7u du 6 So d 7 6 (x + ln x) 7u x 6 7(x + ln x) x b) The differential of secx is sec x tan x (don t forget the s). The sin x requires substitution and the use of chain rule. If y = 6sin x let u = sinx then y = 6u du cosx u du C differentiation Page 5
By chain rule du du cos x u = cos x sin x = 6 sin x So finally: d (6sin x secx) 6 sin x sec x tan x c) The rule for differentiating products is to differentiate the first and times it by the second and then add the differential of the second times by the first. Or in symbols: d(uv) du dv v u d (x tanx) x 0 tanx x sec x A lot of students forget to put the back into the trigonometric function and write sec x. d 5 5x + 5 d) (x e ) Another product to differentiate, remember to differentiate the power on the e and bring it to the front. d (x e ) 5x e 5x e 5 5x + 5 5x + 5 5 5x + 5 5x + 5 5x e ( x) C differentiation Page 6
e) d cos5x x 5 We are asked to differentiate a quotient but this can be rewritten as a product. Most questions can be treated this way unless a question says specifically to use the quotient rule. 5 5 d cos5x d x cos5x x Let u = x v = cos5x 5 du x dv 5x sin 5x 5 Therefore: cos 5x d x 5 x cos 5x 5 x 5x sin5x 5 cos 5x x 5 5x sin5x 5 Example Given that x = 8sin(7y + ), find. The only thing that is new here is that x = 8sin(7y + ) Don t forget to write the (7y + ) in 56cos(7y ) sec(7y ) 56cos(7y ) 56 C differentiation Page 7
Equations of tangents and Normals Example The curve with equation y 6 x e meets the y axis at the point A. a) Prove that the tangent at A to the curve has equation 6y = x + b) The point B has x-coordinate ln 6 and lies on the curve. The normal at B to the curve meets the tangent at A to the curve at the point C. Prove that the x coordinate of C is ln 6 + 7.5 and find the y coordinate of C. It is always best practice to make a sketch of the diagram especially in this case where there are quite a few lines. 6 B Normal y=/6e x C A 0 0 5 Tangent 0 5 0 5 y= 6 ex Normal at x = ln6 a) First find the gradient: e x 6 At A, x = 0 therefore the gradient is 6 as is the y coordinate. y = mx + c C differentiation Page 8
6 = 6 0 + c Therefore y = x 6 6 and hence 6y = x + b) Find the equation of the normal at the point B with x coord of ln 6. From part (a) the gradient function is e x 6 hence the gradient at B is 6 and so the normal gradient is. (y coord also 6) 6 y = mx + c 6 = ln6 6 + c x ln6 y 6 6 6 The normal then meets the tangent from part (a) at the point C. x ln6 x 6 6 6 6 6 x 5 ln6 x 7.5 ln6 7.5 ln 6 Therefore the y coordinate can be found by substituting x = ln 6 + 7.5 into 6y = x +. 6y = ln 6 + 7.5 + y = 6 ln6 7 C differentiation Page 9
Example The curve c has equation 7 y x ln x, where x > 0. The tangent at the point C where x = meets the x axis at the point A. Prove that the x coordinate of A is 9 ln. Once again start by differentiating to find the gradient when x =. 7 y x ln x 5 x x When x grad = y = - ln y = mx + c ln = + c c = -9 ln y = x 9 ln The line meets the x-axis at the point where y = 0. Therefore: x 9 ln = 0 9 ln x C differentiation Page 0
Differentiating Quotients Example 5 Given that x 6x 7 y, x, (x ) Show that 8 (x ) It is obvious from the question that by using the quotient rule it will be easier to get the desired answer. u du dv d v u v v u = x 6x 7 v = (x ) du 8x 6 dv (x ) Therefore: C differentiation Page
(x ) (8x 6) (x 6x 7) (x ) (x ) cancel a factor of (x ) top and bottom (x )(8x 6) 8x x (x ) 8x x 8x x (x ) 8 (x ) Example 6 P The diagram shows part of the curve with equation, y = (0x ) tan x, 0 x < The curve has a minimum at the point P. The x coordinate of P is K. Show that K satisfies the equation 0K 9 + 5 sin 6K = 0 C differentiation Page
As soon as you see the word minimum your first thought should be to differentiate and set it equal to zero. Differentiating a product: y = (0x - ) tan x 0tanx (0x - )sec x At a minimum the gradient is zero 0 0 tanx (0x - )sec x sinx (0x ) 0 = 0 cosx cos x Multiply by cos x 0 0sinxcosx + (0x ) 0 5sin6x + (0x ) If x = K 0K 9 + 5 sin 6K = 0 A lot of the ideas outlined above are not complicated and the final example below deals with turning points and the differential of exponential functions. C differentiation Page
Example 7 a) The curve, C, has equation y x 9 x Use calculus to find the coordinates of the tuning points of C. b) Given that y 5 x e find the value of at the point x = ln. a) The curve C is to be differentiated as a quotient u du dv d v u v v u = x v = 9 + x du dv x 9 x x (9 x ) 9 x (9 x ) C differentiation Page
Turning points exist where = 0 therefore the numerator must equal zero (why not the denominator?). The numerator is the difference of two squares and therefore the values of x must be +/-. By substituting these values into y we get: x y, and, 9 x b) Given that y 5 x e find the value of at the point x = ln. This is a composite function. To differentiate it simply multiply by the power, multiply by the differential of the bracket and then multiply by the differential of the bracket. y e x 5 5 e 5e x x e x e x Let x = ln 5e ln e ln 5 C Differentiation is an area where marks can be scored easily. You should not find these questions difficult! C differentiation Page 5
. (a) Differentiate with respect to x (i) sin x + sec x, (ii) {x + ln (x)}. 5x 0x 9 Given that y =, x, ( x ) Edexcel past examination questions 8 (b) show that =. (6) ( x ) [005 June Q]. The point P lies on the curve with equation y = ln x. The x-coordinate of P is. Find an equation of the normal to the curve at the point P in the form y = ax + b, where a and b are constants. [006 Jan Q]. (a) Differentiate with respect to x (i) x e x +, (ii) cos(x ). x (b) Given that x = sin (y + 6), find in terms of x. [006 Jan Q]. Differentiate, with respect to x, (a) e x + ln x, (b) (5 x ). [006June Q] C differentiation Page 6
5. The curve C has equation x = sin y. (a) Show that the point P, lies on C. () (b) Show that = at P. (c) Find an equation of the normal to C at P. Give your answer in the form y = mx + c, where m and c are exact constants. [007 Jan Q] x 6. (i) The curve C has equation y =. 9 x Use calculus to find the coordinates of the turning points of C. (6) x (ii) Given that y = ( e ), find the value of at x = ln. [007 Jan Q] 7. A curve C has equation y = x e x d y (a) Find, using the product rule for differentiation. (b) Hence find the coordinates of the turning points of C. d y (c) Find. (d) Determine the nature of each turning point of the curve C. 8. A curve C has equation y = e x tan x, x (n + ). () () [007June Q] (a) Show that the turning points on C occur where tan x =. (6) (b) Find an equation of the tangent to C at the point where x = 0. () [008 Jan Q] 9. The point P lies on the curve with equation C differentiation Page 7
y = e x +. The y-coordinate of P is 8. (a) Find, in terms of ln, the x-coordinate of P. () (b) Find the equation of the tangent to the curve at the point P in the form y = ax + b, where a and b are exact constants to be found. [008June Q] 0. (a) Differentiate with respect to x, (i) e x (sin x + cos x), (ii)x ln (5x + ). x 6x 7 Given that y =, x, ( x ) (b) show that 0 ( x ) =. d y d 5 (c) Hence find and the real values of x for which =. y [008June Q6]. (a) Find the value of at the point where x = on the curve with equation y = x (5x ). (6) sin x (b) Differentiate with respect to x. x [009 Jan Q]. Find the equation of the tangent to the curve x = cos (y + ) at 0,. Give your answer in the form y = ax + b, where a and b are constants to be found. (6) [009 Jan Q] C differentiation Page 8
x x. f(x) =. x x x (a) Express f(x) as a single fraction in its simplest form. (b) Hence show that f (x) =. ( x ) [009 Jan Q]. (i) Differentiate with respect to x (a) x cos x, ln( x ) (b). x (ii) A curve C has the equation y = (x+), x >, y > 0. The point P on the curve has x-coordinate. Find an equation of the tangent to C at P in the form ax + by + c = 0, where a, b and c are integers. (6) [009June Q] 5. ln( x ) d y (i) Given that y =, find. x (ii) Given that x = tan y, show that = x. [00 Jan Q] d(sec x) 6. (a) By writing sec x as, show that = sec x tan x. cos x Given that y = e x sec x, d y (b) find. The curve with equation y = e x sec x, 6 < x < 6, has a minimum turning point at (a, b). (c) Find the values of the constants a and b, giving your answers to significant figures. [00 Jan Q7] C differentiation Page 9
7. A curve C has equation y = ( 5 x), 5 x. The point P on C has x-coordinate. Find an equation of the normal to C at P in the form ax + by + c = 0, where a, b and c are integers. (7) [00 June Q] 8. Figure Figure shows a sketch of the curve C with the equation y = (x 5x + )e x. (a) Find the coordinates of the point where C crosses the y-axis. () (b) Show that C crosses the x-axis at x = and find the x-coordinate of the other point where C crosses the x-axis. d y (c) Find. (d) Hence find the exact coordinates of the turning points of C. [00 June Q5] Core Maths Differentiation Page 0
9. The curve C has equation (a) Show that sin x y =. cos x 6sin x cos x ( cos x) = (b) Find an equation of the tangent to C at the point on C where x =. Write your answer in the form y = ax + b, where a and b are exact constants. [0 Jan Q7] 0. Given that (a) show that d (sec x) = sec x tan x. d (cos x) = sin x, Given that x = sec y, (b) find in terms of y. (c) Hence find in terms of x. () [00 Jan Q8]. Differentiate with respect to x (a) ln (x + x + 5), cos x (b). x () [0 June Q] Core Maths Differentiation Page
. f(x) = x 5 (x )( x ) x, x, x. x 9 (a) Show that f(x) = 5. (x )( x ) The curve C has equation y = f (x). The point P 5, lies on C. (b) Find an equation of the normal to C at P. (8) [0 June Q7]. Differentiate with respect to x, giving your answer in its simplest form, (a) x ln (x), sin x (b). x [0 Jan Q]. The point P is the point on the curve x = tan y with y-coordinate. Find an equation of the normal to the curve at P. (7) [0 Jan Q] Core Maths Differentiation Page
5. Figure Figure shows a sketch of the curve C which has equation y = e x sin x, x. (a) Find the x-coordinate of the turning point P on C, for which x > 0. Give your answer as a multiple of. (b) Find an equation of the normal to C at the point where x = 0. (6) [0 June Q] 6. (a) Differentiate with respect to x, (i) x ln (x), 0x (ii) 5, giving your answer in its simplest form. (x ) (b) Given that x = tan y find in terms of x. (6) [0 June Q7] Core Maths Differentiation Page
7. The curve C has equation y = (x ) 5 The point P lies on C and has coordinates (w, ). Find (a) the value of w, () (b) the equation of the tangent to C at the point P in the form y = mx + c, where m and c are constants. [0 Jan Q] 8. (i) Differentiate with respect to x (a) y = x ln x, (b) y = (x + sin x). (6) Given that x = cot y, (ii) show that = x. [0 Jan Q5] Core Maths Differentiation Page
9. h(x) = x + x 5 ( x 8, x 0. 5)( x ) x (a) Show that h(x) =. x 5 (b) Hence, or otherwise, find h (x) in its simplest form. Figure Figure shows a graph of the curve with equation y = h(x). (c) Calculate the range of h(x). 0. Given that x = sec y, 0 < y < 6 [0 Jan Q7] (a) find d x in terms of y. () (b) Hence show that 6 xx ( ) d y (c) Find an expression for in terms of x. Give your answer in its simplest form. [0 June Q5] Core Maths Differentiation Page 5
. Figure shows a sketch of part of the curve with equation y = f(x) where f(x) = (x + x + ) The curve cuts the x-axis at points A and B as shown in Figure. (a) Calculate the x-coordinate of A and the x-coordinate of B, giving your answers to decimal places. () (b) Find f (x). The curve has a minimum turning point P as shown in Figure. (c) Show that the x-coordinate of P is the solution of e x (x ) x ( x ). The curve C has equation y = f (x) where x f ( x) x, x (a) Show that f ( x) 9 x Given that P is a point on C such that f ʹ(x) =, (b) find the coordinates of P. [0_R June Q5] Core Maths Differentiation Page 6
. The curve C has equation x = 8y tan y. [0 June Q] The point P has coordinates, 8. (a) Verify that P lies on C. (b) Find the equation of the tangent to C at P in the form ay = x + b, where the constants a and b are to be found in terms of π. (7). (i) Given that show that (ii) Given that find the exact value of d y at (iii) Given that show that x sec y, 0 f ( x) x x y y x x ln x e x, giving your answer in its simplest form. f ( x) cos x x g( x) x where g(x) is an expression to be found., x, x () [0 June Q] [0_R June Q] Core Maths Differentiation Page 7
5. The point P lies on the curve with equation Given that P has (x, y) coordinates x = (y sin y). p,, where p is a constant, (a) find the exact value of p. () The tangent to the curve at P cuts the y-axis at the point A. (b) Use calculus to find the coordinates of A. (6) [05 June Q5] 6. Given that k is a negative constant and that the function f(x) is defined by ( x 5k)( x k) f (x) =, x 0, x kx k (a) show that f (x) = x k. x k (b) Hence find f ' (x), giving your answer in its simplest form. (c) State, with a reason, whether f (x) is an increasing or a decreasing function. Justify your answer. () x 7. y x 5. [05 June Q9] (a) Find d y, writing your answer as a single fraction in its simplest form. (b) Hence find the set of values of x for which d y < 0. [06 June Q] Core Maths Differentiation Page 8
8. (i) Find, using calculus, the x coordinate of the turning point of the curve with equation y = e x cos x, x. Give your answer to decimal places. (ii) Given x = sin y, 0 < y <, find Write your answer in the form as a function of y. = p cosec(qy), 0 < y <, where p and q are constants to be determined. [06 June Q5] 9. f(x) = x x x 7x 6, x >, x R. x x 6 (a) Given that x x x 7 x 6 B x A x x 6 x, find the values of the constants A and B. (b) Hence or otherwise, using calculus, find an equation of the normal to the curve with equation y = f(x) at the point where x =. [06 June Q6] Core Maths Differentiation Page 9
0. (i) Given y = x(x ) 5, show that (a) = g(x)(x ) where g(x) is a function to be determined. (b) Hence find the set of values of x for which 0 () (ii) Given x = ln(secy), 0 < y < π find. Given that as a function of x in its simplest form. [07 June Q7] show that cos y, sin, d sin where α is a constant to be determined. [0 June, IAL Q]. (a) Use the identity for sin(a + B) to prove that (b) Show that sin A sin Acos A () d ln x cosec x A curve C has the equation y ln tan x sin x, 0 x (c) Find the x coordinates of the points on C where d y 0. Give your answers to decimal places. (6) [0 June, IAL Q0] Core Maths Differentiation Page 0
. Figure shows a sketch of part of the curve C with equation where a is a constant and a > ln. y ax x e e, x The curve C has a turning point P and crosses the x-axis at the point Q as shown in Figure. (a) Find, in terms of a, the coordinates of the point P. (b) Find, in terms of a, the x coordinate of the point Q. (c) Sketch the curve with equation (6) y ax x e e, x, a ln Show on your sketch the exact coordinates, in terms of a, of the points at which the curve meets or cuts the coordinate axes.. The curve C has equation x y, x ( x ) The point P on C has x coordinate. [0 June, IAL Q] Find an equation of the normal to C at the point P in the form ax + by + c = 0, where a, b and c are integers. (6) [05 Jan, IAL Q] Core Maths Differentiation Page
5. Figure shows a sketch of part of the curve with equation y = f(x), where f(x) = (x 5)e x, x The curve has a minimum turning point at A. (a) Use calculus to find the exact coordinates of A. Given that the equation f(x) = k, where k is a constant, has exactly two roots, (b) state the range of possible values of k. (c) Sketch the curve with equation y = f(x). () Indicate clearly on your sketch the coordinates of the points at which the curve crosses or meets the axes. [05 June, IAL Q] Core Maths Differentiation Page
6. gx ( ) x x 7x 8x 8, x >, x R x x (a) Given that find the values of the constants A and B. x x 7x 8x 8 B x A x x x (b) Hence, or otherwise, find the equation of the tangent to the curve with equation y = g(x) at the point where x =. Give your answer in the form y = mx + c, where m and c are constants to be determined. 7. (i) Differentiate y = 5x ln x, x > 0 (ii) Given that y = x sin x + cos x, - p < x < p [06 June, IAL Q] () show that (+ x)sin x + (- x)cos x =, - p + sinx < x < p [07 Jan, IAL Q6] Core Maths Differentiation Page