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Advanced differentiation Calculator

Get detailed solutions to your math problems with our Advanced differentiation 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 advanced differentiation. This solution was automatically generated by our smart calculator:

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

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

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

Apply natural logarithm to both sides of the equality

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

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

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

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

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

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=

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

The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=2\left(-\frac{d}{dx}\left(x\right)\sin\left(x\right)\ln\left(x\right)+\frac{d}{dx}\left(\ln\left(x\right)\right)\cos\left(x\right)\right)$
9

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

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

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=2\left(-\sin\left(x\right)\ln\left(x\right)+\frac{1}{x}\frac{d}{dx}\left(x\right)\cos\left(x\right)\right)$
10

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=2\left(-\sin\left(x\right)\ln\left(x\right)+\frac{1}{x}\frac{d}{dx}\left(x\right)\cos\left(x\right)\right)$
11

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

$\frac{y^{\prime}}{y}=2\left(-\sin\left(x\right)\ln\left(x\right)+\frac{1}{x}\frac{d}{dx}\left(x\right)\cos\left(x\right)\right)$
12

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

$\frac{y^{\prime}}{y}=2\left(-\sin\left(x\right)\ln\left(x\right)+\frac{1}{x}\cos\left(x\right)\right)$

Multiply the fraction by the term

$\frac{y^{\prime}}{y}=2\left(-\sin\left(x\right)\ln\left(x\right)+\frac{1\cos\left(x\right)}{x}\right)$

Any expression multiplied by $1$ is equal to itself

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

Multiply the fraction by the term

$\frac{y^{\prime}}{y}=2\left(-\sin\left(x\right)\ln\left(x\right)+\frac{\cos\left(x\right)}{x}\right)$
14

Combine all terms into a single fraction with $x$ as common denominator

$\frac{y^{\prime}}{y}=2\left(\frac{-x\sin\left(x\right)\ln\left(x\right)+\cos\left(x\right)}{x}\right)$
15

Multiplying the fraction by $2$

$\frac{y^{\prime}}{y}=\frac{2\left(-x\sin\left(x\right)\ln\left(x\right)+\cos\left(x\right)\right)}{x}$
16

Multiply both sides of the equation by $y$

$y^{\prime}=\frac{2y\left(-x\sin\left(x\right)\ln\left(x\right)+\cos\left(x\right)\right)}{x}$
17

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

$y^{\prime}=\frac{2\left(-x\sin\left(x\right)\ln\left(x\right)+\cos\left(x\right)\right)x^{2\cos\left(x\right)}}{x}$
18

Simplify the fraction $\frac{2\left(-x\sin\left(x\right)\ln\left(x\right)+\cos\left(x\right)\right)x^{2\cos\left(x\right)}}{x}$ by $x$

$y^{\prime}=2\left(-x\sin\left(x\right)\ln\left(x\right)+\cos\left(x\right)\right)x^{\left(2\cos\left(x\right)-1\right)}$
19

The derivative of the function results in

$2\left(-x\sin\left(x\right)\ln\left(x\right)+\cos\left(x\right)\right)x^{\left(2\cos\left(x\right)-1\right)}$

Final answer to the problem

$2\left(-x\sin\left(x\right)\ln\left(x\right)+\cos\left(x\right)\right)x^{\left(2\cos\left(x\right)-1\right)}$

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