SOME RULES FOR DERIVATIVES
– compiled by Laura Curley, undergrad content tutor in Math
The Chain Rule
Let F be the composition of two differentiable functions f and g; F(x) = f(g(x)). Then F is differentiable and
F'(x) = f ‘(g(x)) g ‘(x).
Example: Find the derivative of F(x) =(x^3+5x) ^7
Let u=g(x) =x^3+5x and f (u) =u^7.
By the chain rule F'(x) = f ‘(g(x)) g ‘(x)
(7u^6) (3x^2+5)
The last step is to replace u with its original equation:
F'(x) =7(x^3+5x) ^6*(3x^2+5)
The Power Rule
If f(x) = xn where n is a positive integer, then f ‘(x) = n xn –1
Example: Find the derivative of f(x) = x^5
By the product rule f `(x) = 5x^5-1 = 5x^4
The derivative of y=x
The derivative is 1. Example: f(x) = x; f `(x) = 1
Derivative of a Constant
The derivative of y=C, where C is any constant is always 0.
Example 1: f(x)=3: f `(x)=0
Example 2: f(x) =15: f `(x) =0
The Product Rule
h (x) = f(x) g(x)
h`(x) = f `(x) g(x)+ f(x)g `(x)
The derivative according to the product rule is (in words), the derivative of the first function times the second function plus the first function times the derivative of the second function.
Example: f(x) = 5xsin(x):
Let 5x be the first function and sin(x) be the second function.
h`(x) = f `(x) g(x)+ f(x)g `(x)
f ` (x) = (5)sin(x)+5xcos(x)
The Quotient Rule
f(x) = f `(x) g(x)- g`(x) f(x)
g(x) (g(x))^2
An easy way that I remember this is a saying that my class came up with, maybe it will work for you.
(low d hi minus hi d low all over low squared)
The d means to take the derivative of what comes after it.
The top of the fraction is (high).
The bottom of fraction is (low).
Example: f `(x) = 2 = (x+1)(0) – (2)(1) = -2
x+1 (x+1)^2 (x+1)^2
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