HOMEWORK 23 Date: 1. Differentiate the following functions with 8. Differentiate the following functions with respect to X: respect to X: a ) f ( x ) = x4( 1 - 3x) 6 a) y= - cos 2x b ) f ( x ) = 5 5 + sin 2x CLONE Nov 05 V 1 - x 2 b) y = In 5x 1 - X 2 c ) f ( x ) = sin In ( x 2 - 3x ) ] 9 . Find y if y = 2x3 ( cos 3x ) 2 . CLONE dx Oct 06 2. Find y if: dx a) y= _ x + 2 10. Differentiate the following functions with 3x + 5 respect to X: b) y= xe2x + 5x a ) f ( x ) = 3/ x2 - 2 CLONE Oct 08 3. Differentiate the following functions with b ) f ( x ) = In 3X - 2 (cos2 2x respect to X: a) f(x) = sin? (2x + 1) b) f ( x ) = (x2 - 3) (x+ 1)2 dx 11. Find dy if y = tan 1-2x CLONE Apr 09 x2 c ) f ( x ) = - V2 x + 3 12. Find y if y = In- e 2x - 3 CLONE dx 1 - 4x Oct 09 4. Find y if y = x5 + ( x- 2)3 + 10x dx 13. Find y if y = xin (3x- 1) . CLONE dx Apr 11 5. Differentiate the following functions with respect to x: sin X dx 14. Find for the following: a ) f ( x ) = - 1 + cos x b) f ( x ) = (1 + 3x2)(2 + 3x) a) y= - + 2x3 - 3/ x CLONE x 2 Mar 12 c ) f ( x ) = V2 x3 - 5x b) y = x3 \\ x + 3 6. Find if y = dx 5 ( 5 x - 8 ) 3 dx 15. Find y if y = In2 ( 3x + e2x ). CLONE Mar 13 7. Find f'(x) if 16. Find - if y = cos 5-2x a ) f ( x ) = exs- 9x + 7 CLONE dx 5 + 2x Sep 13 b) f( x) = tan x2 + In(4x3 ) ] 23Gradient I Slope of the Tangent A. Find the slope of the tangent line for each of the following functions at the given point. [ml 1. Find the slope of the tangent line for Find the slope of the tangent line for y= 2 1+ 1 when x=2. x f(x) = x2 + l at the point (1, 2). X At the point (1, 2) ' - _;= r(1)2(1) ma 1 Gradient of the tangent, m = 1 2. Find the slope of the tangent line for 3. 'Find the slope of the tangent line for f(x) = X 2 at the point (4, 2). Y = V3'X atthe P0\" (' 1~ 2) X. B. Use the definition of differentiation to determine the slope of tangent to the curve at the given point. rm f(x) =x2 +1 ; (2, 5) f'(x) = [lino W \"m [(x + h)2+ 11(x2+ 1) h>0 h \"m x2+ 2hx + h2 +1x2 1 [140 h 2hx + h2 lim h> 0 = \"m h(2x + h) = lim (2x + h) h> 0 = 2x + (0) = 2X At the oint 2 5 f'(x) = 2(2) = 4 I. Gradient of the tangent, m = 4 24 HOMEWORK 25 Date: Equation of the Tangent Line A. Find the equation of the tangent line to the curve at the given point. Example 1. Find the equation of the tangent line to Find the equation of the tangent line to the the curve y = cos 3x 2x In ( x + 1 ) -2 at X = 0. curve y = - x + 2 at X = 0. Solution Find the y-coordinate When x = 0, y = e 2 (0 ) (0) + 2 2 = (0 , 2 ) Gradient of the tangent: du V - ,dv dy _ dx dx dx V2 u = e2x V = x+2 du dx = e2x(2) av = 1+ 0 dx = 202x = 1 dy = (x + 2)(2e2x ) -e2x(1) dx ( x + 2) 2 = 2e2x ( x + 2) -e2x x + 2) 2 When x = 0 dy _ 2e20) (0 + 2)-e2(0) dx (0 + 2)2 = 3 . Gradient of the tangent, m = Equation of the tangent line: y = mx + C 1 S (0 ) + c 2 2 3 X + 1 2 25e2x 3 +X3 I a) Find the gradient of the tangent to the curve at x = 0. Find the equation to the tangent line Given the function y = b) to the curve at the point [0, g). Given that f(x) = x3 5X + 4. a) Use the definition of derivative to find f'(x). Find the equation of the tangent line to the graph of f(x) at x = 1. b) Find the equation of the tangent line to the curve y = at x = 0. (12x)2 Find the slope of the tangent to the curve y= cos3x atx=0 2In(2x+1) ' CLONE Apr 06 Given that f(x) = 5x x2. a) Use the definition of derivative to find f'(x). b) Find the equation of the tangent line to the graph of f(x) at x = 2. CLONE Oct 06 . , (x2+ sf Given that f(x) = 3e2" a) Find f'(x). b) Find the equation of the tangent line 13_ at x = 0. CLONE Apr 08 Use the definition of differentiation to find the slope of the tangent line for the function f(x) = \\/5x4 at point (4, 4). CLONE Oct 09 26 8. 10. 11. 12. Find the equation of the tangent to the 92" 2x . curve x = In at the ornt f( ) ( 3X + 1 ] p CLONE X ' 0' Oct 10 Find the equation of the tangent line to X2'6 atx=0. '10\" the curve = y 362x Sep 13 Let C be a curve represented by y = . Find an equation of the x + 3 tangent line to C at the point P(3, 0). CLONE Mar 16 Given the function f(x) = m 1. a) Find the gradient of the tangent to the curve at x = 3. b) Find the equation of the tangent line to the curve at the point (3, 2). c) Use the definition of derivative to nd f '(x) for the function. cLouE Oct 16 Given the function f(x) = 1x2 + 12 . a) Find the gradient of the tangent to the curve at x = 2. . b) Find the equation to the tangent line' to the curve at the point (2, 4). c) Use the definition of derivative to nd f'(x) for the function. CLONE Mar 17 . . + x2 . Given the function f(x) = . nd the 8x + 2 equation of the tangent line to the curve CLONE . 1 at the pomt [3, 2') Dec 18 HOMEWORK 21 Date: Derivatives of Exponential, Logarithmic and Trigonometric Functions A. Differentiate each of the following with respect to x. Example 1. y = 6x - ex 2. f ( x ) = Vex f ( x) = - 1 3 x 1 2 -3 x f'(x) = e-3x (- 3) = -3e-3x Example 3. f(x) = 5x + In(1-7x) 4. y = In x 2 ( 1 + x 5x y = In - x2 + 3 1 2 = In (5x) - In (x2+ 3) 1 dy = 1 (5 ) - - 1 dx 5 x x 2 + 3 - (2x) 1 2x = X x2 + 3 Example 5. y = cos(2x - 1) 6. f(x) = sin'(4x + 3) f(x) = cos2 (3x) 2 3 1 = cos (3x) f'(x) = 2 cos(3x) [- sin(3x)] (3) = - 6 sin(3x) cos(3x) = - 3 2 sin(3x) cos(3x) ] = - 3 sin(6x) Example 7. f ( x ) = tan (x2 + 1) 8. y = cot2 (8x) y = sec(1 - x) dy = sec(1 - x) tan(1 - x)(- 1) dx = - sec(1 - x) tan(1 - x) 21HOMEWORK 22 Date: Combination of Differentiation Techniques A. Differentiate each of the following with respect to x. Example 1. y= \\2 + tan(3x) 1 2 f(x) = sin x +1 1 - 3x f'(x) = cos (x + 1 vu' -uv' 1-3x V2 u = x+ 1 V = 1 - 3x u' = 1 V'= - 3 = COS x +1) (1-3x) (1)-(x+1)(-3) 1 - 3x (1-3x)2 = COS x + 1 1-3x + 3x + 3 1-3x (1-3x) 2 = COS x + 1 4 (1-3x ) (1-3x) 2 4 cos x + 1 (1-3x ) 2 1-3x 2. y = cos (x2ex+1) 3. f( x) = x2 tan x + sin(e 2x) 22