# Find the higher order derivative of xcos(x)

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

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$-\sin\left(x\right)-\left(\sin\left(x\right)+x\cos\left(x\right)\right)$

## Step by step solution

Problem

$\frac{d^2}{dx^2}\left(x\cdot \cos\left(x\right)\right)$
1

Rewriting the high order derivative

$\frac{d^{\left(2-1\right)}}{dx^{\left(2-1\right)}}\left(\frac{d}{dx}\left(x\cos\left(x\right)\right)\right)$
2

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

$\frac{d^{\left(2-1\right)}}{dx^{\left(2-1\right)}}\left(x\frac{d}{dx}\left(\cos\left(x\right)\right)+\cos\left(x\right)\frac{d}{dx}\left(x\right)\right)$
3

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

$\frac{d^{\left(2-1\right)}}{dx^{\left(2-1\right)}}\left(x\frac{d}{dx}\left(\cos\left(x\right)\right)+1\cos\left(x\right)\right)$
4

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 $f'(x) = -\sin(x)\cdot D_x(x)$

$\frac{d^{\left(2-1\right)}}{dx^{\left(2-1\right)}}\left(1\cos\left(x\right)-x\sin\left(x\right)\right)$
5

Subtract the values $2$ and $-1$

$\frac{d^{1}}{dx^{1}}\left(1\cos\left(x\right)-x\sin\left(x\right)\right)$
6

Any expression to the power of $1$ is equal to that same expression

$\frac{d}{dx}\left(1\cos\left(x\right)-x\sin\left(x\right)\right)$
7

Any expression multiplied by $1$ is equal to itself

$\frac{d}{dx}\left(\cos\left(x\right)-x\sin\left(x\right)\right)$
8

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

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

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

$\frac{d}{dx}\left(\cos\left(x\right)\right)-\frac{d}{dx}\left(x\sin\left(x\right)\right)$
10

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

$\frac{d}{dx}\left(\cos\left(x\right)\right)-\left(\frac{d}{dx}\left(x\right)\sin\left(x\right)+x\frac{d}{dx}\left(\sin\left(x\right)\right)\right)$
11

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

$\frac{d}{dx}\left(\cos\left(x\right)\right)-\left(1\sin\left(x\right)+x\frac{d}{dx}\left(\sin\left(x\right)\right)\right)$
12

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

$\frac{d}{dx}\left(\cos\left(x\right)\right)-\left(1\sin\left(x\right)+x\cos\left(x\right)\right)$
13

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 $f'(x) = -\sin(x)\cdot D_x(x)$

$-\sin\left(x\right)-\left(1\sin\left(x\right)+x\cos\left(x\right)\right)$
14

Any expression multiplied by $1$ is equal to itself

$-\sin\left(x\right)-\left(\sin\left(x\right)+x\cos\left(x\right)\right)$

$-\sin\left(x\right)-\left(\sin\left(x\right)+x\cos\left(x\right)\right)$

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### Main topic:

Differential calculus

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