Q. 1 - Calculus

Question

Read the section and make your own summary of the material.

Step-by-Step Solution

Answer

When $f\left(x\right)$is an exponential function of the form for some constant 'k'  is given.

When $b\ne 1$ and all appropriate values of 'x' for any constant $b>0$ is given.

Also if $f\left(x\right),f^{-1}\left(x\right)$ are inverse functions and differentiable, then for all appropriate values of 'x' is given.

Step 1. When f x is an exponential function of the form.

Consider the given question,

$f\text{'}\left(x\right)=kf\left(x\right)$ for some constant 'k'  if and only if $f\left(x\right)$ is an exponential function of the form.

$$\begin{gathered} \frac{d}{dx}\left(e^x\right)=e^x \\ \frac{d}{dx}\left(b^x\right)=b^x.\left(\ln b\right) \\ \frac{d}{dx}\left(e^{kx}\right)=k.\left(e^{kx}\right) \end{gathered}$$

Step 2. When b ≠ 1 and all appropriate values of ' x '.

For any constant 'k' any constant $b>0$ with $b\ne 1$ and all appropriate values of 'x'.

$$\begin{gathered} \frac{d}{dx}\left({\log }_b x\right)=\frac{1}{\left(\log b\right)x} \\ \frac{d}{dx}\left(\ln x\right)=\frac{1}{x} \\ \frac{d}{dx}\left(\ln \left|x\right|\right)=\frac{1}{x} \end{gathered}$$

If $f\left(x\right),f^{-1}\left(x\right)$ are inverse functions and differentiable, then for all appropriate values of 'x',

$\left(f^{-1}\right)\text{'}\left(x\right)=\frac{1}{f\text{'}\left(f^{-1}\left(x\right)\right)}$