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Chain Rule and Exponential Function Derivatives
5
Transcript
Chain Rule and Exponential Function Derivatives
Done By: Mouj A.
Definition
A function
The rate of change of a function
The slope of the line tangent to the curve
Chain Rule
Finding the derivative of a composition of two functions.
If y is a function of u and u is a function of x, then y is a function of x.
The chain rule tell us how to find the derivative of y with respect to x
d/dx [ f ( g(x) ) ]
= f ' ( g(x) ) g'(x)
example
Find the derivative of f(x);
f(x)= (x^2 + 3x + 4)^3
f'(x) = 3(x^2 + 2x + 4)^2 (2x + 2 + 0)
= 3(x^2 + 2x + 4)^2 (2x + 2)
= 3(2x + 2)(x^2 + 2x + 4)^2
= (6x + 6)(x^2 + 2x + 4)^2
Exponential Functions
Finding the derivative of natural logarithms.
d/dx e ^f (x)
= f '(x) . e ^f (x)
d/dx a^f (x)
= ( lna ) a ^f (x) . f '(x)
example
Find the derivative of g(x);
g(x) = e ^5x
g'(x) = 5e^5x
Find the derivative of k(x);
k(x) = 2^4x
k'(x) = 2 ^4x ( ln 2 ) (4)
As a rate of change
Of A Derivative
f (x) = x ^2
= 2x
= lim 2x + h
h-> 0
= lim h (2x + h)
h
h-> 0
= lim 2xh + h^2
h
h-> 0
= lim ( x+ h ) ^2 - x ^2
h
h-> 0
f'(x) = lim f (x+h) - f (x)
h
h-> 0
= lim x ^2 +2xh + h^2 - x ^2
h
h-> 0
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