what is the derivative of 2^x?
I know that it's something with ln but I can't find it in my notes
d/dx (a^x) = a^x *ln a
thanks! i just found it in my notes too
To find the derivative of 2^x, you can use differentiation rules. In this case, the chain rule is applied because 2^x is in the form of a composite function.
Let's start by expressing 2^x as e^(x * ln(2)). Now we can apply the chain rule.
The chain rule states that if we have a composite function f(g(x)), where f and g are both differentiable functions, then the derivative of f(g(x)) is given by f'(g(x)) * g'(x).
In this case, g(x) = x * ln(2) and f(u) = e^u. Therefore, we need to find f'(u) and g'(x).
The derivative of f(u) = e^u is simply f'(u) = e^u.
The derivative of g(x) = x * ln(2) can be found using the product rule, which states that if we have two functions u(x) and v(x), then the derivative of u(x) * v(x) is given by u'(x) * v(x) + u(x) * v'(x). Here, u(x) = x and v(x) = ln(2).
Differentiating u(x) = x gives us u'(x) = 1.
Differentiating v(x) = ln(2) gives us v'(x) = 0, since ln(2) is a constant.
Now we can substitute these values into the chain rule formula:
f'(g(x)) = e^(x * ln(2)) * 1 = e^(x * ln(2)) = 2^x
So, the derivative of 2^x is 2^x.