Differentiate $a^{x}$ w.r.t. $x$, where $a$ is a positive constant.

Step 1: Rewrite the expression using base e

To differentiate $a^x$, it's often helpful to rewrite it using the natural exponential function $e$. We know that any positive number $N$ can be written as $e^{\ln N}$. Applying this property, we can express $a^x$ as $e^{\ln(a^x)}$.

Step 2: Simplify the exponent

Using the logarithm property $\ln(M^P) = P \ln M$, we can simplify the exponent. Here, $M = a$ and $P = x$, so $\ln(a^x)$ becomes $x \ln a$.

Step 3: Differentiate using the chain rule

Now we differentiate $e^{x \ln a}$ with respect to $x$. We use the chain rule, which states that $\frac{d}{dx}(e^{f(x)}) = e^{f(x)} \cdot f'(x)$. Here, $f(x) = x \ln a$.

Step 4: Differentiate the exponent

Since $a$ is a positive constant, $\ln a$ is also a constant. The derivative of $x$ multiplied by a constant is simply that constant. Therefore, the derivative of $x \ln a$ with respect to $x$ is $\ln a$.

Step 5: Substitute back and simplify

Substitute the derivative of the exponent back into the chain rule result. We know that $e^{x \ln a}$ is equivalent to $a^x$. Therefore, the derivative of $a^x$ is $a^x \ln a$.

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