Mastering The Chain Rule: Derivatives Of Composite Functions Explained With Example Of F(X)^N

Chain rule of f(x)^n

n(f(x)^(n-1))f'(x)

The chain rule is a fundamental rule in calculus that allows us to find the derivative of a composite function. If we have a function f(x) and another function g(x) where g(x) is a function of f(x), then the chain rule tells us that the derivative of g(x) with respect to x is:

(g(f(x)))’ = g'(f(x)) * f'(x)

This means that we take the derivative of g(x) with respect to f(x) and then multiply it by the derivative of f(x) with respect to x.

Now, let’s consider the function f(x)^n, where n is a constant. We can rewrite this function as (f(x))^n, which is a composite function where g(u) = u^n and f(x) is the input to g. Using the chain rule, we can find the derivative of f(x)^n as follows:

((f(x))^n)’ = (g(f(x)))’ = g'(f(x)) * f'(x)

where g(u) = u^n and f(x) is the input to g. To find g'(u), we can use the power rule, which tells us that if g(u) = u^n, then g'(u) = n*u^(n-1). Substituting this into the chain rule formula, we get:

((f(x))^n)’ = g'(f(x)) * f'(x) = n*(f(x))^(n-1) * f'(x)

Therefore, the derivative of f(x)^n is n*(f(x))^(n-1) times the derivative of f(x) with respect to x. This rule can be useful in computing derivatives of functions that involve powers, such as polynomials and trigonometric functions.

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