power rule proof

The power rule is a basic rule in calculus for finding the derivative of a function of the form f(x) = x^n, where n is a constant. It states that:

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

To prove the power rule, we will use the definition of the derivative:

f'(x) = lim(h -> 0) [(f(x + h) - f(x))/h]

Let f(x) = x^n . We want to find the derivative f'(x). So, we plug in our function into the definition of the derivative:

f'(x) = lim(h -> 0) [((x + h)^n - x^n)/h]

Next, we use the binomial theorem to expand (x + h)^n:

(x + h)^n = C(n,0) * x^n * h^0 + C(n,1) * x^(n-1) * h^1 + ... + C(n,n) * x^0 * h^n

where C(n,k) denotes the binomial coefficient, which is equal to n!/(k! * (n-k)!).

Now substitute this back into our limit expression for f'(x):

f'(x) = lim(h -> 0) [ (C(n,0) * x^n * h^0 + C(n,1) * x^(n-1) * h^1 + ... + C(n,n) * x^0 * h^n - x^n) / h]

Next, we can simplify by dividing the expression by h:

f'(x) = lim(h -> 0) [ C(n,1) * x^(n-1) + C(n,2) * x^(n-2) * h + ... + C(n,n) * x^0 * h^(n-1)]

Now, let's take the limit as h approaches 0:

f'(x) = C(n,1) * x^(n-1)

Finally, note that C(n,1) = n:

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

And that's our power rule. So, for any function of the form x^n, the derivative is given by:

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