Question

Find the derivative of y=a^x

Answer 1

If y = aˣ, then y = exp(ln(aˣ)) = exp(x ln(a)). (Here I use exp(x) = eˣ.)

Then using the chain rule, the derivative of y is

dy/dx = exp(x ln(a)) • ln(a)

and undoing the rewriting, we end up with

dy/dx = ln(a) aˣ

Answer 2

The derivative of y = a^x is ln(a) * a^x. Substitute specific values for 'a' to obtain the numerical result.

To find the derivative of the function y = a^x, where a is a constant, you can use the chain rule. The chain rule states that if you have a composite function f(g(x)), then the derivative is given by f'(g(x)) * g'(x). In this case, let f(u) = a^u and g(x) = x.

The derivative of f(u) = a^u with respect to u is ln(a) * a^u, and the derivative of g(x) = x with respect to x is 1. Applying the chain rule, we get:

dy/dx = f'(g(x)) * g'(x) = ln(a) * a^x * 1 = ln(a) * a^x

Therefore, the derivative of y = a^x is ln(a) * a^x. If you have any specific values for a, you can substitute them in to get the numerical result.