poetinjam4gj4

2023-03-26

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

iistiidgllj

Beginner2023-03-27Added 6 answers

Take both sides' natural logarithms.

$\mathrm{ln}\left(f\left(x\right)\right)=\mathrm{ln}\left({5}^{\mathrm{ln}x}\right)$

Using the following rule, the right side can be made simpler: $\mathrm{log}\left({a}^{b}\right)=b\mathrm{log}\left(a\right)$

$\mathrm{ln}\left(f\left(x\right)\right)=\mathrm{ln}\left(x\right)\mathrm{ln}\left(5\right)$

Distinguish the opposing sides. Remember that the left-hand side will employ the chain rule. Remember that $\mathrm{ln}\left(5\right)$ is just a constant and will remain on the right-hand side.

$\frac{1}{f\left(x\right)}\cdot f\prime \left(x\right)=\frac{1}{x}\cdot \mathrm{ln}\left(5\right)$

To solve for f'(x), the derivative, multiply both sides by f(x).

$f\prime \left(x\right)=\frac{\mathrm{ln}\left(5\right)}{x}\cdot f\left(x\right)$

Rewrite f(x) as $5}^{\mathrm{ln}x$.

$f\prime \left(x\right)=\frac{{5}^{\mathrm{ln}x}\mathrm{ln}5}{x}$

$\mathrm{ln}\left(f\left(x\right)\right)=\mathrm{ln}\left({5}^{\mathrm{ln}x}\right)$

Using the following rule, the right side can be made simpler: $\mathrm{log}\left({a}^{b}\right)=b\mathrm{log}\left(a\right)$

$\mathrm{ln}\left(f\left(x\right)\right)=\mathrm{ln}\left(x\right)\mathrm{ln}\left(5\right)$

Distinguish the opposing sides. Remember that the left-hand side will employ the chain rule. Remember that $\mathrm{ln}\left(5\right)$ is just a constant and will remain on the right-hand side.

$\frac{1}{f\left(x\right)}\cdot f\prime \left(x\right)=\frac{1}{x}\cdot \mathrm{ln}\left(5\right)$

To solve for f'(x), the derivative, multiply both sides by f(x).

$f\prime \left(x\right)=\frac{\mathrm{ln}\left(5\right)}{x}\cdot f\left(x\right)$

Rewrite f(x) as $5}^{\mathrm{ln}x$.

$f\prime \left(x\right)=\frac{{5}^{\mathrm{ln}x}\mathrm{ln}5}{x}$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?

What is the limit as x approaches negative infinity of $x+\sqrt{{x}^{2}+2x}$?