d/dx (x^n)
nx^n-1 (power rule)
The derivative of x^n with respect to x is nx^(n-1). This can be proven using the power rule of differentiation, which states that the derivative of x^n with respect to x is nx^(n-1). Here’s how to derive this formula:
Power Rule Derivation:
Let f(x) = x^n be a function of x.
Using the definition of the derivative, we have:
f'(x) = lim(h->0) [f(x+h) – f(x)] / h
= lim(h->0) [(x+h)^n – x^n] / h
Expanding (x+h)^n using the binomial theorem, we get:
f'(x) = lim(h->0) [x^n + nx^(n-1)h + (terms with h^2 and higher powers of h)] / h – x^n
= lim(h->0) [nx^(n-1)h + (terms with h^2 and higher powers of h)] / h
Canceling the x^n term that appears in both the numerator and denominator, and taking the limit as h approaches 0, the higher-order terms with h^2 and higher powers of h vanish, leaving:
f'(x) = nx^(n-1)
Thus, the derivative of x^n with respect to x is nx^(n-1)
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