LU8 Solutions for Examples.ppt - LU8 Derivative I The Derivative Four Basic Rules of Differentiation The General Power Rule The Product and Quotient

LU8 Solutions for Examples.ppt - LU8 Derivative I The...

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Unformatted text preview: LU8 Derivative I The Derivative Four Basic Rules of Differentiation The General Power Rule The Product and Quotient Rules: Higher-Order Derivatives The Chain Rule Differentiation of Exponential and Logarithmic Functions Rule 2: The Power Rule Practice Examples 2: Find the derivative of f ( x ) x Rule 2: The Power Rule Practice Examples 2 solution: Find the derivative of f ( x ) x d f ( x ) dx d 1/2 x x dx 1 1/2 1 x 2 1 2 x 1 1/2 x 2 Rule 2: The Power Rule Practice Examples 3: 1 Find the derivative of f ( x ) 3 x Rule 2: The Power Rule Practice Examples 3 solution: 1 Find the derivative of f ( x ) 3 x f ( x ) d 1 d 1/3 x 3 dx x dx 1 1/3 1 x 3 1 4/3 1 x 4 / 3 3 3x Rule 3: Derivative of a Constant Multiple Function Practice Examples 5: 3 Find the derivative of f ( x ) x Rule 3: Derivative of a Constant Multiple Function Practice Examples 5: 3 Find the derivative of f ( x ) x f ( x) d 3 x 1/ 2 dx 1 3/ 2 3 x 2 3 3/ 2 2x Rule 4: The Sum Rule Practice Examples 7: Find the derivative of t2 5 g (t ) 3 5 t Rule 4: The Sum Rule Practice Examples 7: Find the derivative of t2 5 g (t ) 3 5 t d t2 5 d 1 2 g (t ) 3 t 5t 3 dt 5 t dt 5 1 d 2 d 3 t 5 t 5 dt dt 1 2t 5 3t 4 5 2t 15 2t 5 75 4 5 t 5t 4 The General Power Rule Practice Examples 11: Find the derivative of f ( x ) 1 4x 2 7 2 The General Power Rule Practice Examples 11: Find the derivative of f ( x ) 1 4x 2 7 2 Solution 2 Rewrite as a power function: f ( x ) 4 x 2 7 Apply the general power rule: 2 f ( x ) 2 4 x 7 16 x 4x 2 7 3 3 8x Rule 5: The Product Rule Practice Examples 14: Find the derivative of f ( x ) x 3 x 1 Rule 5: The Product Rule Practice Examples 14: Find the derivative of f ( x ) x 3 x 1 d 1/2 d 3 1/2 f ( x ) x x 1 x 1 x dx dx 3 1 1/2 x x x1/2 1 3x 2 2 3 1 x 5/2 3x 5/2 3x 2 2 7 x 5/2 3x 2 2 Rule 5: The Product Rule Practice Examples 10: 5 Find the derivative of f ( x ) x 2 2 x 3 Rule 5: The Product Rule Practice Examples 10: 5 Find the derivative of f ( x ) x 2 2 x 3 Solution Apply the product rule and the general power rule: f ( x ) x 2 d 5 5 d 2 x 3 2 x 3 x 2 dx dx 2 x 5 2 x 3 4 2 2 x 3 2 x 4 10 x 2 2 x 3 2 x 2 x 3 4 2 x 2 x 3 5 x 2 x 3 2 x 2 x 3 4 5 7 x 3 5 Rule 6: The Quotient Rule Practice Examples 16: Find the derivative of x2 1 f ( x) 2 x 1 Rule 6: The Quotient Rule Practice Examples 16: Find the derivative of x2 1 f ( x) 2 x 1 d 2 d 2 2 x 1 dx x 1 x 1 dx x 1 f ( x ) 2 2 x 1 2 x 2 1 2 x x 2 1 2 x x 2 1 2 2 x3 2 x 2 x3 2 x x 2 1 2 4x x 2 1 2 Rule 6: The Quotient Rule Practice Examples 12: Find the derivative of 2x 1 f ( x ) 3 x 2 3 Rule 6: The Quotient Rule Practice Examples 12: 3 2x 1 Find the derivative of f ( x ) 3 x 2 Solution Apply the general power rule and the quotient rule: 2 2x 1 d 2x 1 f ( x ) 3 3 x 2 dx 3 x 2 2x 1 3 3 x 2 2 2 3x 2 2 2 x 1 3 2 3x 2 2x 1 6x 4 6x 3 2 3x 2 3x 2 2 3 3 2 x 1 4 3 x 2 Higher-Order Derivatives Practice Examples 19: Find the second derivative of the function f(x) = (2x2 +3)3/2 Higher-Order Derivatives Practice Examples 19: Find the second derivative of the function f(x) = (2x2 +3)3/2 Solution Using the product rule we get the second derivative: 1/2 1/2 d d 2 2 f ( x ) 6 x 2 x 3 2 x 3 6 x dx dx 1/2 1/2 1 2 2 6 x 2 x 3 4 x 2 x 3 6 2 2 2 12 x 2 x 3 2 6 2 x 3 6 4 x 2 3 2 x2 3 1/2 1/2 2 6 2 x 3 1/2 2 x 2 2 x 2 3 Examples 23 Find the derivative of the function g (t ) et 2 3/2 Examples 23 Find the derivative of the function g (t ) et 2 Solution Using the general power rule gives 1/2 d 3 t g (t ) e 2 et 2 2 dt 1/2 t 3 t e 2 e 2 1/2 3 t t e e 2 2 3/2 Examples 24 Find the derivative of the function f ( x ) e 2 x 3x 3x Find the derivative of the function y e 2 t 2 t Find the derivative of the function g (t ) e 2x Find the derivative of the function y xe 2 x et Find the derivative of the function g (t ) t e e t Examples 24 Find the derivative of the function f ( x ) e 2 x Solution d f ( x ) e 2x dx e 2 x (2) 2x 2e 2 x Examples 24 Find the derivative of the function y e 3x Solution dy 3x d e ( 3x ) dx dx e 3 x ( 3) 3e 3 x Examples 24 2 Find the derivative of the function g (t ) e 2 t Solution d g (t ) e 2t 2 t dt 2 t 2 t (4t 1)e 2 t 2 t t Examples 24 Find the derivative of the function y xe 2 x Solution dy d 2x 2x d x e e x dx dx dx d x e 2 x 2 x e 2 x (1) dx xe 2 x ( 2) e 2 x 2 xe 2 x e 2 x e 2 x (1 2 x ) Examples 24 t e Find the derivative of the function g (t ) et e t Solution t t d t t d t t e e e e e e dt dt g (t ) t t 2 e e t t t t t t e e e e e e e e t t 2 e2t 1 e2t 1 t t 2 e e 2 t t 2 e e Examples 26 ln x x Find the derivative of the function g ( x ) ln x Solution Examples 26 ln x x Find the derivative of the function g ( x ) ln x Solution d d x (ln x ) ln x ( x ) dx g ( x ) dx x2 1 x ln x (1) x 2 x 1 ln x x2 ...
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