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Higher-order derivatives Calculator

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1

Solved example of higher-order derivatives

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

Apply the rule of the cube of a binomial

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

Calculate the power ${\left(-1\right)}^2$

$\left(x-3\right)\frac{d^2}{dx^2}\left(\left(x-1\right)\left(x^2-2x+1\right)\right)$
4

Multiplying polynomials $x$ and $-2x+2$

$\left(x-3\right)\frac{d^2}{dx^2}\left(x^{3}-x^2-2x^2+2x+x-1\right)$
5

Adding $2x$ and $x$

$\left(x-3\right)\frac{d^2}{dx^2}\left(3x+x^{3}-x^2-2x^2-1\right)$
6

Adding $-1x^2$ and $-2x^2$

$\left(x-3\right)\frac{d^2}{dx^2}\left(-3x^2+3x+x^{3}-1\right)$
7

Rewriting the high order derivative

$\left(x-3\right)\frac{d^{\left(2-1\right)}}{dx^{\left(2-1\right)}}\left(\frac{d}{dx}\left(-3x^2+3x+x^{3}-1\right)\right)$
8

Subtract the values $2$ and $-1$

$\left(x-3\right)\frac{d^{1}}{dx^{1}}\left(\frac{d}{dx}\left(-3x^2+3x+x^{3}-1\right)\right)$
9

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

$\left(x-3\right)\frac{d}{dx}\left(\frac{d}{dx}\left(-3x^2+3x+x^{3}-1\right)\right)$
10

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

$\left(x-3\right)\left(\frac{d}{dx}\left(\frac{d}{dx}\left(-3x^2\right)\right)+\frac{d}{dx}\left(\frac{d}{dx}\left(3x\right)\right)+\frac{d}{dx}\left(\frac{d}{dx}\left(x^{3}\right)\right)+\frac{d}{dx}\left(\frac{d}{dx}\left(-1\right)\right)\right)$
11

The derivative of the constant function ($\frac{d}{dx}\left(-1\right)$) is equal to zero

$\left(x-3\right)\left(\frac{d}{dx}\left(\frac{d}{dx}\left(-3x^2\right)\right)+\frac{d}{dx}\left(\frac{d}{dx}\left(3x\right)\right)+\frac{d}{dx}\left(\frac{d}{dx}\left(x^{3}\right)\right)\right)$
12

The derivative of the linear function times a constant, is equal to the constant

$\left(x-3\right)\left(\frac{d}{dx}\left(\frac{d}{dx}\left(-3x^2\right)\right)+\frac{d}{dx}\left(3\right)+\frac{d}{dx}\left(\frac{d}{dx}\left(x^{3}\right)\right)\right)$
13

The derivative of the constant function ($3$) is equal to zero

$\left(x-3\right)\left(\frac{d}{dx}\left(\frac{d}{dx}\left(-3x^2\right)\right)+\frac{d}{dx}\left(\frac{d}{dx}\left(x^{3}\right)\right)\right)$
14

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

$\left(x-3\right)\left(-3\frac{d}{dx}\left(\frac{d}{dx}\left(x^2\right)\right)+\frac{d}{dx}\left(\frac{d}{dx}\left(x^{3}\right)\right)\right)$
15

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

$\left(x-3\right)\left(-3\frac{d}{dx}\left(2x^{1}\right)+\frac{d}{dx}\left(3x^{2}\right)\right)$
16

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

$\left(x-3\right)\left(-3\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(3x^{2}\right)\right)$
17

The derivative of the linear function times a constant, is equal to the constant

$\left(x-3\right)\left(-3\cdot 2+\frac{d}{dx}\left(3x^{2}\right)\right)$
18

Multiply $-3$ times $2$

$\left(x-3\right)\left(-6+\frac{d}{dx}\left(3x^{2}\right)\right)$
19

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

$\left(x-3\right)\left(-6+3\frac{d}{dx}\left(x^{2}\right)\right)$
20

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

$\left(-6+6x\right)\left(x-3\right)$
21

Multiplying polynomials $-3$ and $-6+6x$

$-6x+6x^2+18-18x$
22

Adding $-6x$ and $-18x$

$-24x+6x^2+18$

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