Derivative of a^u: Understanding the Chain Rule for Exponential Functions

derivative of a^u

To find the derivative of a function of the form a^u, where a is a constant and u is a function of x, we can use the chain rule

To find the derivative of a function of the form a^u, where a is a constant and u is a function of x, we can use the chain rule.

The chain rule states that if we have a function of the form f(g(x)), then the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x).

In the case of a^u, let’s first write it as f(g(x)) = a^u(x), where f(x) = a^x and g(x) = u(x).

Now let’s find the derivative of f(x) = a^x with respect to x. This derivative is well-known and can be written as f'(x) = ln(a) * a^x. So, the derivative of f(g(x)) = a^u(x) with respect to x is f'(g(x)) * g'(x).

We can now find g'(x), which is the derivative of u(x). Depending on the complexity of u(x), finding g'(x) may involve different techniques (e.g. product rule, chain rule, etc.).

Once you find g'(x), multiply it with f'(g(x)). The resulting expression will be the derivative of a^u(x) with respect to x.

Let’s illustrate this with an example:

Suppose we want to find the derivative of 2^x^2. Here, a = 2 and u(x) = x^2.

First, find f'(x) = ln(2) * 2^x (derivative of 2^x with respect to x).

Next, find g'(x) = derivative of x^2 with respect to x = 2x.

Now, multiply f'(g(x)) = ln(2) * 2^(x^2) with g'(x) = 2x.

The final result is d/dx(2^x^2) = ln(2) * 2^(x^2) * 2x = 2^(x^2 + 1) * ln(2) * x.

So, the derivative of a^u(x) = 2^x^2 is 2^(x^2 + 1) * ln(2) * x.

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