Understanding the Chain Rule: Calculating the Derivative of f(x)^n

Chain rule of f(x)^n

The chain rule is a fundamental rule in calculus that allows us to find the derivative of composite functions

The chain rule is a fundamental rule in calculus that allows us to find the derivative of composite functions. In the case of the chain rule for a function raised to a power, we can apply it to find the derivative of f(x)^n.

Let’s assume we have a function f(x) raised to the power of n, where n is a constant. You can think of it as f(x) being the base function and n being the exponent.

To find the derivative of f(x)^n, we need to differentiate the base function f(x) and then multiply it by the derivative of the exponent n.

The derivative of the base function f(x) is represented as f'(x), and the derivative of the exponent n is represented as d/dx(n).

So, mathematically, the chain rule for f(x)^n can be written as:

(d/dx)[f(x)^n] = n * f(x)^(n-1) * f'(x)

Let’s break down the steps and explain the reasoning:

1. Take the derivative of the base function f(x) using f'(x). This gives us df(x)/dx.

2. Take the derivative of the exponent n with respect to x, represented as d/dx(n). Since n is a constant, the derivative of a constant with respect to x is 0. Therefore, d/dx(n) = 0.

3. Multiply df(x)/dx by d/dx(n). Since d/dx(n) = 0, the derivative of the exponent doesn’t affect the result.

4. Multiply the result by n and multiply it with f(x) raised to the power of (n-1).

So, the final result is n * f(x)^(n-1) * f'(x), which gives us the derivative of f(x)^n.

It’s important to note that for the chain rule to be applicable, both the base function f(x) and the exponent n must be differentiable functions. Additionally, if the base function f(x) is composed of multiple functions, you may need to apply the chain rule multiple times, depending on the complexity of the expression.

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