Question

Explain how the formula for differentiating the natural exponential function is a special case of the formula for differentiating exponential functions of the form $e^{kx}$. Then explain why it is a special case of the formula for differentiating functions of the form $b^x$.

Short answer

For $e^{kx}$we take k and e as constants so it is a form of$e^{ax}$

For $b^x$we take b=e because it is a special case of the form$e^x$

Step-by-step solution

. Given Information

We are given two forms :$e^{kx}and{b}^x$

Step 2. Differentiating ekx 

We differentiate the form :

$\frac{d}{dx}{e}^{kx}=k{e}^{kx},kandearecons\tan ts$

. Differentiating bx

We differentiate the form :

$\frac{d}{dx}{b}^x=b^x{\log }_e\left(b\right)$

We use b=e because it is a special case, so we get :

$\frac{d}{dx}{e}^x=e^x{\log }_e\left(e\right)=e^x$

. Explaining

So we can say the $e^{kx}$is a form of $e^{ax}$and $b^x$is a special form of$e^x$