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Derivative of Logarithmic Functions Calculator

Get detailed solutions to your math problems with our Derivative of Logarithmic Functions step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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Example

Solved Problems

Difficult Problems

1

Here, we show you a step-by-step solved example of derivative of logarithmic functions. This solution was automatically generated by our smart calculator:

$\frac{d}{dx}\left(x^{x+2}\right)$
2

To derive the function $x^{\left(x+2\right)}$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=x^{\left(x+2\right)}$
3

Apply natural logarithm to both sides of the equality

$\ln\left(y\right)=\ln\left(x^{\left(x+2\right)}\right)$
4

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$\ln\left(y\right)=\left(x+2\right)\ln\left(x\right)$
5

Derive both sides of the equality with respect to $x$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(\left(x+2\right)\ln\left(x\right)\right)$
6

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x+2$ and $g=\ln\left(x\right)$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(x+2\right)\ln\left(x\right)+\left(x+2\right)\frac{d}{dx}\left(\ln\left(x\right)\right)$

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(x+2\right)\ln\left(x\right)+\left(x+2\right)\frac{1}{x}\frac{d}{dx}\left(x\right)$
7

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(x+2\right)\ln\left(x\right)+\left(x+2\right)\frac{1}{x}\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$\frac{y^{\prime}}{y}=\frac{d}{dx}\left(x+2\right)\ln\left(x\right)+\left(x+2\right)\frac{1}{x}$
8

The derivative of the linear function is equal to $1$

$\frac{y^{\prime}}{y}=\frac{d}{dx}\left(x+2\right)\ln\left(x\right)+\left(x+2\right)\frac{1}{x}$

Multiply the fraction by the term

$\frac{y^{\prime}}{y}=\frac{d}{dx}\left(x+2\right)\ln\left(x\right)+\frac{1\left(x+2\right)}{x}$

Any expression multiplied by $1$ is equal to itself

$\frac{y^{\prime}}{y}=\frac{d}{dx}\left(x+2\right)\ln\left(x\right)+\frac{x+2}{x}$
9

Multiply the fraction by the term

$\frac{y^{\prime}}{y}=\frac{d}{dx}\left(x+2\right)\ln\left(x\right)+\frac{x+2}{x}$

The derivative of the constant function ($2$) is equal to zero

$\frac{y^{\prime}}{y}=\frac{d}{dx}\left(x\right)\ln\left(x\right)+\frac{x+2}{x}$

The derivative of the linear function is equal to $1$

$\frac{y^{\prime}}{y}=\ln\left(x\right)+\frac{x+2}{x}$
10

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{y^{\prime}}{y}=\ln\left(x\right)+\frac{x+2}{x}$
11

Multiply both sides of the equation by $y$

$y^{\prime}=\left(\ln\left(x\right)+\frac{x+2}{x}\right)y$
12

Substitute $y$ for the original function: $x^{\left(x+2\right)}$

$y^{\prime}=\left(\ln\left(x\right)+\frac{x+2}{x}\right)x^{\left(x+2\right)}$
13

The derivative of the function results in

$\left(\ln\left(x\right)+\frac{x+2}{x}\right)x^{\left(x+2\right)}$

Final answer to the problem

$\left(\ln\left(x\right)+\frac{x+2}{x}\right)x^{\left(x+2\right)}$

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