Chain rule of f(x)^n
n(f(x)^(n-1))f'(x)
The chain rule is a method for finding the derivative of a composite function. In the case of the function f(x)^n, we need to apply the chain rule in combination with the power rule.
The chain rule states that for a composite function f(g(x)), the derivative of the function is given by the product of the derivative of the outer function and the derivative of the inner function, or:
(d/dx) f(g(x)) = f'(g(x)) * g'(x)
Applying the chain rule to f(x)^n, we first recognize that this function is a composite of two functions: f(x) and g(x) = x^n.
So, we can write f(x)^n = (f(g(x)))^n as a composite function. Using the chain rule, the derivative of f(x)^n is:
(d/dx) f(x)^n = (d/dx) (f(g(x)))^n
= n * (f(g(x)))^(n-1) * (d/dx) (f(g(x))) // Power rule
= n * (f(g(x)))^(n-1) * f'(g(x)) * (d/dx) g(x) // Chain rule
= n * f(x)^(n-1) * f'(x) * nx^(n-1) // Substitute g(x) with x^n
= n * f(x)^(n-1) * f'(x) * n(x^(n-1)) // Simplify
Therefore, the formula for the derivative of f(x)^n is:
(d/dx) f(x)^n = n * f(x)^(n-1) * f'(x) * n(x^(n-1))
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