9.1 Derivative of a function 9.3 Differentiation of exponential, logarithmic and trigonometric functions
OBJECTIVES b. discuss the differentiability of a function at x = a 9.1 Derivative of a function (a) find the derivative of a function f(x) using the first principle ′() = → ( + ) − () At the end of the lesson students are able to:
9.3 Differentiation of exponential, logarithmic and trigonometric functions (a) Find the derivatives of the functions. (b) Apply the basic rule of differentiation for the following function where u=g(x) is any function of x. (c) Solve problems involving the combination of differentiation rules. OBJECTIVES
DERIVATIVE OF A FUNCTION The derivative of a function y = f (x) with respect to x can be denoted as dy dx y ' f x'( ) [ ( )] d f x dx
FIND DERIVATIVE BY THE FIRST PRINCIPLE By using the first principle, the derivative of a function with respect to is given by 0 ( ) ( ) '( ) limh f x h f x f x h Find the derivative of the following functions with respect to x using the first principle 2 a f x x ) ( ) b f x x ) ( ) 1 Example 1
2 f x x ( ) 0 ( ) ( )'( ) limh f x h f xf x h h x hx h x f x h 2 2 2 0 2 '( ) lim h hx h h 2 0 2 lim h h x h h (2 ) lim 0 lim (2 ) 0 x h h 2x 2 2 2 f x h x h x hx h ( ) ( ) 2 Solution (a)
0 1 1 '( ) limh x h x f x h f x x ( ) 1 0 1 lim 1 h x h x h 1 11 1x h xx h x 0 1 lim 1 1 1 h x h x h x x h f x h x h ( ) 1 0 ( ) ( )'( ) limh f x h f xf x h Solution (b)
0 lim 1 1 h h h x h x 0 1 lim 1 1 h x h x 1 x x 1 1 2 1 1 x
Differentiability OF A FUNCTION A function f is differentiable at = if exists, i.e. ( ) ( ) '( ) lim x a f x f a f a x a ( ) ( ) ( ) ( ) lim lim x a x a f x f a f x f a x a x a
NOTE 1. If a function f is differentiable at = , then f is continuous at = . 2. There are function that are continuous but not differentiable. 3. If function is not differentiable then continuity of the functions at = must be tested.
Given Is f differentiable at x = 0? 2 2 3 , 0( ) . 3 , 0x x f x x x Remarks: Calculate f (0) using f (x) = x 2 + 3 Example 2
Solution 2 f 0 0 3 3 2 0 3 lim 0 3 x x x 0 ( ) (0) lim x 0 f x f x 2 0 lim x x x 0 lim x x 0 0 2 3 lim 0 3x x x 0 ( ) (0) lim x 0 f x f x 0 2 lim x x x 0 lim 2 x 2 2 2 3 , 0( ) .3 , 0x xf x x x 0 0 ( ) (0) ( ) (0) Since lim lim , x x 0 0 f x f f x f x x therefore ′(0) does not exist and hence f is not differentiable at x = 0.
A function f is defined by Determine the value of k if f is differentiable for all real values. 2 1, 1 ( ) . 1 , 1 x x f x k x x Example 3 Remarks: f is differentiable for all real values f is differentiable at x=1 and therefore 1 1 1 1 lim lim x x 1 1 f x f f x f x x
1 1 lim lim 1 1 1 1 x x f x f x x x f f 2 f 1 1 1 0 2 1 1 1 1 lim lim 1 0 0 x x 1 x k x x x Solution 1 1 1 1 1 lim lim x x 1 1 x x k x x x 1 1 lim 1 lim x x x k 1 1 k k 2
1. Derivatives of Constant Function 0 dy dx BASIC OF DIFFERENTIATION ( ) ( ) 5 a f x ( ) '( ) 0 a f x Example 1 Solution ( ) ( ) 2 b f x ( ) '( ) 0 b f x y k k R , 2(c) ( ) f x e(c) '( ) 0f x
( ) for n f x x dy n 1 n x dx , n y x n R 3 (a) ( ) f x x 2 ( ) '( ) 3 a f x x 3 (b) ( ) 5f x x 2( ) '( ) 15b f x x Example 2 Solution BASIC OF DIFFERENTIATION 2. Derivatives of function n x
( ) f x ii y e . ( ) x i y e ' dy f x f x edx BASIC OF DIFFERENTIATION 3. Derivatives of Exponential Functions dy x e dx
. 3. Derivatives of Exponential Functions ( ) ( ) f x iv y a For 0 & 1 a a ( ) '( ) lndy f x f x a adx lndy x a adx ( ) x iii y a
2 3 4 ( ) x x b y e Find the derivative with respect to for the following: x ( ) ( ) 4 x a f x e Example 3 ( ) 5x a y 2 2 ( ) 3x x b y
2 2 3 43 4 dy d x xx x e dx dx ' ' x xf x f xd e f x edx d e f x edx 2 3 4 6 4 dy x x x e dx Solution (a) '( ) 4 x f x e Solution (b) 2 3 4 x x y e ( ) 4 x f x e
( ) ( ) ln '( ) lnx x f x f x d a a adx d a f x a adx Solution (a) dx dy 5 ln 5 x Solution (b) 2 23 ln 3 2 2 x x x dx dy 2 13 ln 3 2 2 x x x 2 2 2 2 3 ln3dy d x x x x dx dx 5x y 2 2 3 x x y
4. Derivatives of Logarithmic Functions ( ) ln i y x ii y f x ln ( ) . dy 1 dx x '( )( )dy f xdx f x for 0 x for ( ) 0 f x BASIC OF DIFFERENTIATION
(a) ln3 (b) ( ) ln 4 1 (c) ln 1 1 (d) ln 1 y x f x x y x x x f x x Example 4 Differentiate the following with respect to x
y x ln3 dy dx 3 3x 1 x Solution (a) Alternative method y x ln3 ln3 ln x dy 1 dx x '( )ln ( ) ( )d f xf x dx f x
f x 4 1 4 2 1 x 4 1 2 x f x x ( ) ln 4 1 Solution (b) 1 2 ln 4 1 x 1 ln 4 1 2 x '( )ln ( ) ( )d f xf x dx f x
y x x ln 1 1 1 2 1 dy dx x x Solution (c) '( )ln ( ) ( )d f xf x dx f xy x x ln ln 1 1 ln ln 1 2 y x x
1 1 1 1 f x x x 1 ( ) ln 1 x f x x Solution (b) '( )ln ( ) ( )d f xf x dx f xf x x x ( ) ln 1 ln 1 1 1 1 1 f x x x
The differentiation of trigonometric functions are only true for angles which are measured in radians. 5. Derivatives of Trigonometric Functions x x dx d sin cos cos sin d x x dx x x dx d 2 tan sec x x dx d 2 cot cosec sec sec tan d x x xdx cosec cosec cot d x x xdx BASIC OF DIFFERENTIATION
5. Derivatives of Trigonometric Functions ' sin cos d f x f x f x dx cos sin ' d f x f x f x dx 2 tan sec ' d f x f x f x dx The differentiation of trigonometric functions are ONLY true for angles which are measured in radians. cosec cosec cot ' d f x f x f x f x dx sec sec tan ' d f x f x f x f x dx 2 cot cosec ' d f x f x f x dx
dy dx 2cos x3sin x 2cosx3sinx Solution sin coscos sind x xdx d x xdx Differentiate with respect to .y 2sin 3cos x x xExample 5
2 ' sec ln ln d f x x x dx Solution Differentiate with respect to .f x x =tan ln xExample 6 2 tan sec 'd f x f x f xdx 2 sec ln ' x f x x
Solution 2 2 2 cosec cot 2 2 2 dy x x d x dx dx 2 2 cosec cot 2 2 dy x x x dx 2 2 cosec cot 2 2 x x x cosec cosec cot ' d f x f x f x f xdx 2 cosec 2 Differentiate with respect to . xx y Example 7
CONCLUSION Derivative of a function f using first principle: h f x h f x f x h ( ) ( ) ( ) lim 0 ' Differentiability of a function at x = a: If a function f is differentiable at x0 , then f is continuous at x0 . ( ) ( ) '( ) lim x a f x f a f a x a
2. i) If () = , then ′() = −1 1. If () = , then f’(x) = 0 CONCLUSION Derivatives of a constant function. Derivatives of function n x ii) If () = , then ′() = −1
d x x e e dx ' d f x f x e f x e dx Derivatives of Exponential Functions ln d x x a a a dx ( ) ( ) '( ) ln d f x f x a f x a a dx CONCLUSION for 0 & 1 a a for 0 & 1 a a
1 ln d x dx x '( ) ln ( ) ( ) d f x f x dx f x Derivatives of Logarithmic Functions CONCLUSION for 0 x for ( ) 0 f x
Derivatives of Trigonometric Functions CONCLUSION Angles are measured in radians. x x dx d sin cos x x dx d cos sin x x dx d 2 tan sec x x dx d 2 cot cosec x x x dx d sec sec tan x x x dx d cosec cosec cot
Derivatives of Trigonometric Functions CONCLUSION 2 tan sec ' d f x f x f x dx cosec cosec cot ' d f x f x f x f x dx sec sec tan ' d f x f x f x f x dx 2 cot cosec ' d f x f x f x dx ' sin cos d f x f x f x dx cos sin ' d f x f x f x dx
0 ( ) ( )'( ) limh f x h f xf x h 3 0 3 '( ) lim 3 3 h f x h x h x 0 3 lim h h h 0 lim 3 h f x h x h x h( ) 3( ) 3 3 Solution (a) Exercise 1 By using the first principle, find ′() for derivative of the following functions : a) ( ) 3 f x x 1 b) ( ) 1 f x x a) ( ) 3 f x x
0 1 1 1 lim h 1 1 x x h h x h x 0 1 1 '( ) lim 1 1 1 h x h x h x f 1 ( ) 1 f x x 1 ( ) 1 f x h x h 0 0 ( ) ( )'( ) lim1 lim ( ) ( )h h f x h f xf x hf x h f xh Solution (b)
0 lim h ( 1) 1 h h x h x 0 1 lim h x h x 1 1 1 x x 1 1 2 1 x 1
3 ( ) ( ) x a f x e 2 ( ) ( ) x b f x e ( ) ( ) x c f x e Exercise (Exponential Function) 1. Find the derivative with respect to for the following: x
3 '( ) 3 x f x e 2 '( ) 2 x f x xe 1 2 '( ) 2 x x f x e 2 x e x Solution 1 (a) Solution 1 (b) Solution 1 (c)
Exercise (Logarithmic Function) 1. Differentiate the following with respect to . x 2 2 2 (a) ( ) ln , 2 1 x g x x x 1 (b) ln 1 x h x x 2 (c) ln x y x e 2 2 2 2. If ln , prove that (4 1) 0. 2 1 x d y dy y x x dx dx
2 1 2 2 g(x) ln (x 2) ln (x 1) ln ( 1) 2 1 2 ln 2 2 x x 2 1 1 2 ' 2 2 2 1 x g x x x 2 1 2 2 x x x Solution 1(a)
h x x x ln 1 ln 1 1 1 1 1 h x x x 1 2 2 x Solution 1(b)
dy dx 2 ln x y x e 2 ln ln x x e 2ln x x 1 2 1 1 2 2 x x 2 1 x 2 x Solution 1(c)
y x x ln ln(2 1) 1 2 2 1 dy dx x x 2 1 2 (2 1) dy x x dx x x 2 2 1 1 2 (2 ) dy dx x x dy x x dx Solution 2 2 2 2 2 2 (2 ) (4 1) (4 1) d y x x x dx dy x dx 2 2 2 (4 1) 0 d y dy x dx dx
Exercise (Trigonometric Function) Differentiate the following with respect to . x 2 sin (a) tan 3 6 5 (b) 4tan 5 (c) sin ln (d) (e) ln (cot ) x y x y x x y x y e y x 2 2 2 3 1 tan (f) ln 1 tan (g) sec(2 3)(h) cosec ( 1) (i) sin cos x y x y x x y x y x x
Solution (a) Solution (b) 2 sec 3 3 6 dy x dx 2 3sec 3 6 x Solution (c) dy dx 2 2 4sec 5 x x 2 2 5 4sec x x dy dx x x 1 cos ln x cos ln x