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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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- Fall '19
- rita atake