Outline. MS121: IT Mathematics. Differentiation Rules for Differentiation: Part 1. Outline. Dublin City University 4 The Quotient Rule

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1 MS2: IT Mathematics Differentiation Rules for Differentiation: Part John Carroll School of Mathematical Sciences Dublin City University Pattern Observe You may have notice the following pattern when we were ifferentiating from first principles: x x 2 2x x 3 3x 2 x x 2 x 2 2 x 3 Differentiation (2/5) MS2: IT Mathematics John Carroll 3 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 4 / 36

2 General Rule Example To fin the erivative of x, note that x = x 2 can be applie with n = 2 to obtain so that the general rule x x n = n x n an this rule is true for all values of n x x 2 = 2 x 2 = 2 x 2 = 2 x Note This rule is foun on page 25 of the formulae an tables booklet, along with a number of other useful erivatives which you may use without proof unless you have been explicitly aske to ifferentiate from first principles Differentiation (2/5) MS2: IT Mathematics John Carroll 5 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 6 / 36 The Exponential an Log Functions Other Entries in the Mathematical Tables The erivatives of the trigonometric functions are also available, for example sin x x = cos x cos x x = sin x x tan x = sec2 x Note that trigonometric efinitions are on pages 3 6 Rate of Growth Another important erivative foun in the log tables is for the exponential function We know that the erivative of a function is equal to its slope which we also think of as being its rate of growth Differentiation (2/5) MS2: IT Mathematics John Carroll 7 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 8 / 36

3 The Exponential an Log Functions The function y = e x Question What function has a erivative equal to the function itself? Answer The answer is the unique function the exponential function For this function only, y = e x = exp(x) y x = y, ie x ex = e x Differentiation (2/5) MS2: IT Mathematics John Carroll 9 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 0 / 36 y = e x an y x = ex The Exponential an Log Functions The Exponential Function (Cont ) Note that e x is simply the number which we call e raise to the power of x (e = e = ) This number, like the number π, is an irrational number, ie a non-repeating ecimal Since 2 < e < 3, the function e x satisfies 2 x < e x < 3 x, an the limit of e x as x must be infinite, ie lim x ex =, Differentiation (2/5) MS2: IT Mathematics John Carroll / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 2 / 36

4 The Exponential an Log Functions The Exponential an Log Functions The Exponential Function (Cont ) The limit of e x as x is lim x ex = lim z e z = lim z e z = = 0 In the foregoing, we simply mae the substitution z = x The Logarithmic Function The function y = e x has an inverse, namely the natural logarithm of x y = ln x, A graph of y = ln x will show that the function is only efine on (0, ) which is the range of the exponential function y = e x an hence the omain of y = ln x The erivative of ln x is also in the tables: x ln x = x Differentiation (2/5) MS2: IT Mathematics John Carroll 3 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 4 / 36 The function y = ln x y = e x is a reflection of y = ln x in y = x Differentiation (2/5) MS2: IT Mathematics John Carroll 5 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 6 / 36

5 Sums & Differences of Functions Work Plan We now introuce some rules for ifferentiation which will allow us to take the erivative of sums, proucts, quotients an compositions of functions In the remainer of this section, we eal with sums of functions while the proucts, quotients an compositions of functions are ealt with in separate sections to follow Differentiation (2/5) MS2: IT Mathematics John Carroll 7 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 8 / 36 Sums & Differences of Functions Derivative of the Sum The erivative of the sum is simply the sum of the erivatives: Example For y = x 2 + 7, we obtain x [ x ] = x x = 2x + 0 = 2x x x for any 2 functions of x [u(x) + v(x)] = u x + v x Note how the erivative of any constant term is zero You can prove this from first principles or simply apply the general for x n with n = 0, ie x 7 = x 7x 0 = 7 x x 0 = 7 0 x = 0 Differentiation (2/5) MS2: IT Mathematics John Carroll 9 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 20 / 36

6 The Prouct Rule Example 2 Using the same rule, we fin x [ex + ln x] = x ex + x ln x = ex + x Differentiation (2/5) MS2: IT Mathematics John Carroll 2 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 22 / 36 The Prouct Rule The Prouct Rule The Prouct Rule Formula To ifferentiate the prouct of two functions, u(x) an v(x), we must use the prouct rule, which is given in the Math Tables: x [u(x) v(x)] = v u x + u v x Example 3 Consier the function y = xe x To use the prouct rule, let u = x an v = e x so that u x =, v x = ex The prouct rule then gives: v u x + u v x = e x + x e x = e x ( + x) Differentiation (2/5) MS2: IT Mathematics John Carroll 23 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 24 / 36

7 The Prouct Rule The Prouct Rule Example 4 For the function y = ln x tan x, we let so that u x = x, The prouct rule then gives: v u x + u v x u = ln x, v = tan x, v x = sec2 x = tan x x + ln x sec2 x = x tan x + ln x sec2 x Example 5 ( Consier y = x x + x 2 ) With we obtain u x = 4 x 3 4, The prouct rule then gives: v u x + u v x u = x 4, v = 2 + 3x + x 2, v x = 3 + 2x = ( 2 + 3x + x 2) 4 x x 4 (3 + 2x) = 4 x ( x + x 2 ) + x 4 (3 + 2x) Differentiation (2/5) MS2: IT Mathematics John Carroll 25 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 26 / 36 The Prouct Rule The Prouct Rule Extension of the Formula If you nee to fin the erivative of 3 functions, say u(x), v(x) an w(x) multiplie together, then the formula to use is an extension of the prouct rule, namely x u v w [u(x) v(x) w(x)] = x vw + u x w + uv x This rule is not foun in the Math Tables because it is simply the prouct rule applie twice Example 6 Differentiate y = e x sin x tan x We let so that u = e x, v = sin x, w = tan x, u x = ex, v x = cos x, w x = sec2 x Differentiation (2/5) MS2: IT Mathematics John Carroll 27 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 28 / 36

8 The Prouct Rule y = e x sin x tan x Example 6 (Cont ) We then obtain x [uvw] = u x v w vw + u w + uv x x = e x sin x tan x + e x cos x tan x + e x sin x sec 2 x = e x { sin x tan x + cos x tan x + sin x sec 2 x } = e x sin x { tan x + + sec 2 x } Differentiation (2/5) MS2: IT Mathematics John Carroll 29 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 30 / 36 Formula To ifferentiate the quotient of two functions, u(x) an v(x), namely y = u(x) v(x), we must use the quotient rule, which is given in the log tables: Example 7 Fin y x where y = x 4 cos x With x [ ] u(x) = v u u v x x v(x) v 2 u x u = x 4 v = cos x = 4 x 3 4 v u x u v x v 2 = cos x 4 x 3 4 x 4 ( sin x) 4 = x 3 4 cos x + x 4 sin x cos 2 x v x = sin x Differentiation (2/5) MS2: IT Mathematics John Carroll 3 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 32 / 36

9 Example 8 Fin y x where y = x 2 + x 2 ln x We let u = x 2 + x 2, v = ln x, so that u x = 2 x 2 2 x 3 v 2, x = x The quotient rule then gives: v u x u v x v 2 = ln x 2 ( ) ( ) x 2 x 3 2 x 2 + x 2 x ln 2 x Example 8 Cont The quotient rule: v u x u v x v 2 = = y = x 2 + x 2 ln x ( ) ( ) ln x 2 x 2 x 3 2 x 2 + x 2 x ln 2 x ( ) 2 ln x x 2 x 3 2 x ln 2 x ( ) x 2 + x 2 Differentiation (2/5) MS2: IT Mathematics John Carroll 33 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 34 / 36 Example 9 Consier the function y = sin x cos x In this case, we have so that u v = cos x, x The quotient rule then gives u = sin x, v = cos x, x = sin x Example 9 Cont The quotient rule: y = sin x cos x v u x u v x cos x cos x sin x ( sin x) v 2 = = cos2 x + sin 2 x = v u x u v x v 2 = cos x cos x sin x ( sin x) = sec 2 x Note that this is just the result x tan x = sec2 x Differentiation (2/5) MS2: IT Mathematics John Carroll 35 / 36 Differentiation (2/5) MS2: IT Mathematics John Carroll 36 / 36