Finding the generalized formula for derivative of power functions
Consider a power function y = f(x) = xn. So lets find the derivative of power function. Let us perform log on both sides, log y = log (xn) = n log x. We know that derivative of log x is equal to 1/x. So derivative on both side gives 1/y * dy/dx = n / x => dy/dx = ny / x. But y = xn. Substituting this value in above equation, dy/dx = n xn / x = nxn-1. This formula can be used for finding derivatives of power functions with any powers. Similarly it can be derived that the derivative of a power function f(x) = x-n is -nxn-1. The derivative power function formula also holds true for irrational values of n. The generic formula for y = a xn is given by y' = an xn-1.
Examples for finding derivative of power functions:
The following examples will help to have a detailed knowledge regarding the derivative of power functions.
(1) Find the derivative of y = 4x8:
Comparing with the standard y = axn, a=4 and n=8. Hence the derivative is given by y' = 4 * 8x7.
Thus y' = 32x7.
(2) Find the derivative of √x:
Here y = √x = x1/2. Applying the standard formula for derivation, y = ½ * x-1/2 = 1/2√x.
(3) Is y = 4x a power function. If so find the derivative?
y = 4x can also be written as y = 4x1. This is a linear power function with power 1. hence it's derivative can be found as y' = 4 * 1 x1-1 = 4.
(4) Find the derivative of x3:
Comparing with y = axn, we have a = 1 and n = 3. Substituting in y' = a xn-1, y' = 3 x2.
(5) Find the function f(x) if f'(x) = 2x:
We have f'(x) = anxn-1 for f(x) = axn, n=2 and a =1. Hence the power function is f(x) = x2.
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