Exercise 14

Implicit differentiation enables us a different perspective from which to see why the rule $\frac{d}{dx} [a^x] = a^x \ln(a)$ holds, if we assume that $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$. This exercise leads you through the key steps to do so.

1. Let $y = a^x$. Rewrite this equation using the natural logarithm function to write $x$ in terms of $y$ (and the constant $a$).

2. Differentiate both sides of the equation you found in (a) with respect to $x$, keeping in mind that $y$ is implicitly a function of $x$.

3. Solve the equation you found in (b) for $\frac{dy}{dx}$, and then use the definition of $y$ to write $\frac{dy}{dx}$ solely in terms of $x$. What have you found?