A-Level Mathematics DIFFERENTIATION I G. David Boswell - Math Camp Typeset 1.1 SET C Review ~ If a and n are real constants, then! DIFFERENTIATION OF POLYNOMIALS Problems ~ Find the first derivative of the following functions w.r.t. the independent variable. (Note the use of the different but equivalent notations.) 1.! y = x 2 + 2x + ; =? d ( ) axn = nax n 1, n! 2.! y = x 5 4x + 9x 6 ; =?.! f (x) = x 9 5x 8 + x +12 ; f '(x) =? 4.! y(x) = 1 4 x8 1 6 x6 x + 2 ; =? 5.! y = 1 x + 1 x 1 2 x ; y'(x) =? 6.! y = x 1 x 2 + 2 x ; y' =? 7.! y = x + 1 ; x =? 8.! y = 2 x + 4 x 2 9.! y = x2 16 + 2 x x 2 + 1 x 2 + x 10.! f (x) = x + 5 ; df =? 11.! y = x +1 x 2 12.! y = 2x 4 1.! y = 14.! y = x2 + 2x +1 5x + 4 x 2 + 2 HAMPTON SCHOOL G. David BÖ ZïK Inc. EPSE / MSE Reference (subject to change) Page 1 of 8
SET D DIFFERENTIATION USING THE CHAIN RULE Review ~ If y and x are functions such that! y = f (u) and! x = g(u), then by the Chain Rule,!. This can also be written as!. Furthermore, note that!. = du du = du du = 1 Problems ~ With an appropriate substitution, use the Chain Rule to find the first derivative of the following functions w.r.t. the independent variable.!! 1. y = sin x ; 14. y = cos8x ; u = x u = 8x 15.! 16.! y = sin( 1x + 2) ; u = 1x + 2 y = 4 cos( 8x 2 + 4x +1) ; u = 8x 2 + 4x +1 17.! sin x 18.! 19.! 20.! cos x 22.! x + 1 x x 2 +1 1 x + 2, x 2 21.!( 9x + 6x 2 10x +1) 5 2.!( x + 2) 4 24.!( 4x 2 + 2x) 2 25.! x 2 + 1 x 1 26.! ( 1 x 2 ) 4 9 (By the end of this exercise, you should be able to solve many of these problems by inspection. (Class discussion: what is meant by an inner derivative?)) HAMPTON SCHOOL G. David BÖ ZïK Inc. EPSE / MSE Reference (subject to change) Page 2 of 8
! SET E DIFFERENTIATION USING THE PRODUCT RULE Review ~ If! u(x) and! v(x) are differentiable functions of! x, then, the derivative of the product! y(x) = u(x)v(x) can be found using the formula = d(uv) = u dv + v du This is often times abbreviated as! d(uv) = udv + vdu or! (uv)' = uv'+ vu' Problems ~ With appropriate substitutions and without expanding, use the Product Rule to find the first derivative of the following functions w.r.t. the independent variable. 27.! y = ( 2x +1) ( x 2) ; =? 28.! 29.! y = 50( 5 x 2 )( 2x + 7) ; y' =? y = ( x 2 + 5) ( 1 x) 0.!( x 2 +1) 1 x 1.!( x +1)sin x ( ) hint: with y = f (x) then find 2.!( x 2 + x 1)cos x.! 4.! 5.! y = x( x x) y = ( x +1) ( x 1) y = 2x 5x + 4 hint: 1 5x + 4 = ( 5x + 4) 1 6.! y = 4x 2 1 ( + x 1) x 1 x 2 HAMPTON SCHOOL G. David BÖ ZïK Inc. EPSE / MSE Reference (subject to change) Page of 8
! SET F DIFFERENTIATION USING THE QUOTIENT RULE Review ~ If! u(x) and! v(x) are differentiable functions with respect to! x, and importantly,! v(x) 0, then the derivative of the quotient! y(x) = u(x) can be found using the formula v(x) = d u v = v du u dv v 2 vdu udv vu' uv' This is often times abbreviated as! d(u / v) = or! (u / v)' =. v 2 v 2 Problems ~ With appropriate substitutions, use the Quotient Rule to find the first derivative of the following functions w.r.t. the independent variable. Also, state the condition for which the function and its derivative exist. 7.! y = x2 +1 x 1 ; =? 8.! 9.! y = x +1 x 2 ; y = x2 9x +1 x +1 =? ; y'(x) =? 40.! y = ( x 2) ; y' =? x 2 41.! g(x) = x4 100 2e x ; dg =? 42.! 4.! y = tan x hint: tan x = sin x cos x y = cot x hint: tan x = cos x sin x 44.! y = sec x hint:? 45.! y = csc x hint:? HAMPTON SCHOOL G. David BÖ ZïK Inc. EPSE / MSE Reference (subject to change) Page 4 of 8
! SET G INTRO TO THE APPLICATIONS OF DIFFERENTIATION Recap / Review ~ First Principles:! f '(x) = lim h 0 f (x + h) f (x) h d dv Product Rule:! (uv) = u i.e.,! + v du (uv)' = uv'+ vu' d u v du Quotient Rule:! i.e.,! v = u dv, v 0 (u / v)' = v 2 vu' uv' v 2, v 0 Chain Rule:! = du du Reciprocal :! such that! = 1 = du du Small Change: Using the first principle definition of the derivative of a function, we write! such that when! is small,!. = lim Δy Δx Δx 0 Δx Δy Δx update is Hence,! Δy. So, overall, given that!, the Δx xnew = x old + Δx y new = y old + Δy y old + Δx HAMPTON SCHOOL G. David BÖ ZïK Inc. EPSE / MSE Reference (subject to change) Page 5 of 8
Problems ~ Solve the following problems, some of which are parametric and rate of change equations. 46. Find the coordinates of the point on the curve! y = 6x 2 + 2x + 5 where the gradient or slope vanished! (hint: find! and solve! for! ). = 0 x 47. A circle! C is defined by the pair of parametric equations given as! x = 5 + cosθ and! y = 8 + sinθ. Find the following: (a)! and! dθ dθ (b) Hence or otherwise,! (c) Coordinates of the point! P(x, y) on the circle where! θ = π c (d) Gradient at the! P(x, y) on! C where! θ = π c 2 48. Given that! g(t) = 1 and! f (t) = 2t, then t 2 2 dg df (a) Find! and! dt dt df (b) Hence or otherwise,! in terms of! t (hint: use the chain rule) dg (c) Using the result in (b), find an expression for! dg df 49. Provided that a curve is described by the equation! y = u 4u 2 + 6u where! u = x 2 +1, then, find the value of! y'(2). (Hint:! y'(2) = ). x=2 HAMPTON SCHOOL G. David BÖ ZïK Inc. EPSE / MSE Reference (subject to change) Page 6 of 8
50. Use the chain rule to find an expression for! given that! y = 5t 2 + t 1 where! t = 4x +10. 51. A curve is parametrically defined by! y = 5t 2 and! x = t +1. Find the gradient of the point on the curve where t = 2. 52. If the cartesian coordinates of a certain trace are uniquely determined at any time by the equations! x = t and! y = sint, find an expression in terms of t for!.! cm s 1!!!!! 5. Bonus ~ A large helium filled balloon at an exciting birthday party is losing gas at a constant rate of.0. A (ner) observer, instead of dancing, wishes to find the rate at which the radius is decreasing when its surface area is 400π cm 2. (a) dv What is the value of the rate of change in volume,? dt (b) Compute the radius of the balloon when its surface area is 400π cm 2. (c) What is the rate at which the radius is decreasing when its surface area has reached the value stated above? (Hints: Assume the balloon is spherical of radius r such that such that its volume is! V = 4 and surface area is! ) πr A = 4πr 2 54. Bonus ~ If water is being pumped from the base of a large upright cylindrical reservoir at a rate of 100 m /min, find the rate at which the height of the water in the reservoir is changing if its diameter is 10 m long. (Hints: (i) Write down the formula for the volume of a cylinder, (ii) note the constants, and (iii) differentiate both sides with respect to the variable, time! t ) HAMPTON SCHOOL G. David BÖ ZïK Inc. EPSE / MSE Reference (subject to change) Page 7 of 8
55. Bonus ~ Given that the square root function is expressed by the equation! y = x, using small changes principle, estimate the value of! 16.005. - ENFIN - HAMPTON SCHOOL G. David BÖ ZïK Inc. EPSE / MSE Reference (subject to change) Page 8 of 8