Exploring the Chain Rule in Calculus | Derivative of a^u with respect to x

d/dx(a^u)

To find the derivative of the function f(x) = a^u with respect to x, where a is a constant and u is a function of x, we can use the chain rule from calculus

To find the derivative of the function f(x) = a^u with respect to x, where a is a constant and u is a function of x, we can use the chain rule from calculus. The chain rule allows us to find the derivative of a composite function.

Let’s begin by writing f(x) = a^u as f(x) = e^(u * ln(a)), where e is Euler’s number and ln(a) is the natural logarithm of a.

Now, let’s find the derivative of f(x) using the chain rule. The chain rule states that if we have a composite function g(u) and u is a function of x, then the derivative of g(u) with respect to x is given by g'(u) * u’.

In our case, g(u) = e^u, and u = u(x). So we need to find the derivative of g(u) with respect to u (which is g'(u)), and then multiply it by the derivative of u with respect to x (which is u’).

The derivative of g(u) = e^u with respect to u is simply g'(u) = e^u.

The derivative of u with respect to x, denoted as u’, is found using the chain rule again. If u = u(x), then u’ = du(x)/dx.

Finally, we can put all the pieces together:

d/dx(a^u) = d/dx(e^(u * ln(a))) (using f(x) = e^(u * ln(a)))

= e^(u * ln(a)) * d/dx(u * ln(a)) (using the chain rule)

= a^u * (u’ * ln(a)) (since d/dx(u) = u’)

= a^u * u’ * ln(a)

So, the derivative of f(x) = a^u with respect to x is a^u * u’ * ln(a).

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