Q. 5 - Calculus

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 $b^x$ .

Step-by-Step Solution

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 1 . Given Information

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

Step 2. Differentiating e k x

We differentiate the form :

$\frac{d}{dx} e^{kx}=k{e}^{kx} ,k and e are cons\tan ts$

Step 3 . Differentiating b x

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 :

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

Step 4 . Explaining

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