Mastering the Power Rule: Derivatives of x^n With Respect to x Explained

d/dx (x^n)

nx^n-1

The derivative of x^n with respect to x is given by:

d/dx(x^n) = n*x^(n-1)

Where n is a constant.

To understand why this is the case, we can use the power rule of differentiation, which states that:

d/dx(x^n) = n*x^(n-1)

This rule can be proven using the limit definition of a derivative, which involves taking the limit as h approaches 0 of the following expression:

(x+h)^n – x^n
————–
h

Simplifying this expression using the binomial theorem, we get:

x^n + nx^(n-1)*h + (other terms in h)

Taking the limit as h approaches 0, we see that the only term that does not go to 0 is the one involving x^(n-1), which has a coefficient of n. Therefore, the derivative of x^n with respect to x is n*x^(n-1).

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