# Derive the function x(x^2+1) with respect to x

## \frac{d}{dx}\left(x\cdot \left(x^2+1\right)\right)

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ln
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sin
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asin
acos
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asinh
acosh
atanh
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$1+3x^2$

## Step by step solution

Problem

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=1+x^2$

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

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

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

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

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

The derivative of the constant function is equal to zero

$x\left(0+\frac{d}{dx}\left(x^2\right)\right)+1\left(1+x^2\right)$
5

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$x\left(0+2x\right)+1\left(1+x^2\right)$
6

$x+0=x$, where $x$ is any expression

$2x\cdot x+1\left(1+x^2\right)$
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Any expression multiplied by $1$ is equal to itself

$2x\cdot x+1+x^2$
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When multiplying exponents with same base you can add the exponents

$2x^2+1+x^2$
9

Adding $2x^2$ and $x^2$

$1+3x^2$

$1+3x^2$

### Main topic:

Differential calculus

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