Understanding the Chain Rule of Differentiation for f(x) = a^x

d/dx a^x

To find the derivative of the function f(x) = a^x with respect to x, where a is a constant, you can apply the chain rule of differentiation

To find the derivative of the function f(x) = a^x with respect to x, where a is a constant, you can apply the chain rule of differentiation. The chain rule states that if you have a composite function, such as f(g(x)), where f and g are both differentiable functions, then the derivative of the composite function with respect to x can be found by multiplying the derivative of the outer function f'(g(x)) with the derivative of the inner function g'(x).

In this case, f(x) = a^x can be written as the composite function f(g(x)), where f(u) = a^u and g(x) = x. Therefore, to find the derivative, we need to consider the derivative of the outer function f'(u) and the derivative of the inner function g'(x).

The derivative of the outer function f(u) = a^u can be obtained using the natural logarithm. The derivative of a^u with respect to u is given by:

f'(u) = (ln a) * a^u.

Here, (ln a) represents the natural logarithm of a.

Now, we need to find the derivative of the inner function g'(x) = 1, since g(x) = x. This is because the derivative of x with respect to x is always 1.

Finally, we can use the chain rule to find the derivative of f(x) = a^x:

f'(x) = f'(u) * g'(x)
= (ln a) * a^u * 1
= (ln a) * a^x.

Therefore, the derivative of f(x) = a^x with respect to x is f'(x) = (ln a) * a^x.

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