# Sum rule of differentiation Calculator

## Get detailed solutions to your math problems with our Sum rule of 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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### Difficult Problems

1

Solved example of sum rule of differentiation

$\frac{d}{dx}\left(4x^3+9x^2-4x-5\right)$
2

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

$\frac{d}{dx}\left(4x^3\right)+\frac{d}{dx}\left(9x^2-4x-5\right)$
3

The derivative of a function multiplied by a constant ($4$) is equal to the constant times the derivative of the function

$4\frac{d}{dx}\left(x^3\right)+\frac{d}{dx}\left(9x^2-4x-5\right)$

$4\cdot 3x^{\left(3-1\right)}+\frac{d}{dx}\left(9x^2-4x-5\right)$

Subtract the values $3$ and $-1$

$4\cdot 3x^{2}+\frac{d}{dx}\left(9x^2-4x-5\right)$

Multiply $4$ times $3$

$12x^{2}+\frac{d}{dx}\left(9x^2-4x-5\right)$
4

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

$12x^{2}+\frac{d}{dx}\left(9x^2-4x-5\right)$
5

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

$12x^{2}+\frac{d}{dx}\left(9x^2\right)+\frac{d}{dx}\left(-4x-5\right)$
6

The derivative of a function multiplied by a constant ($9$) is equal to the constant times the derivative of the function

$12x^{2}+9\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-4x-5\right)$

$4\cdot 3x^{\left(3-1\right)}+\frac{d}{dx}\left(9x^2-4x-5\right)$

Subtract the values $3$ and $-1$

$4\cdot 3x^{2}+\frac{d}{dx}\left(9x^2-4x-5\right)$

Multiply $4$ times $3$

$12x^{2}+\frac{d}{dx}\left(9x^2-4x-5\right)$

$12x^{2}+9\cdot 2x^{\left(2-1\right)}+\frac{d}{dx}\left(-4x-5\right)$

Subtract the values $2$ and $-1$

$12x^{2}+9\cdot 2x^{1}+\frac{d}{dx}\left(-4x-5\right)$

Multiply $9$ times $2$

$12x^{2}+18x^{1}+\frac{d}{dx}\left(-4x-5\right)$

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

$12x^{2}+18x+\frac{d}{dx}\left(-4x-5\right)$
7

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

$12x^{2}+18x+\frac{d}{dx}\left(-4x-5\right)$
8

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

$12x^{2}+18x+\frac{d}{dx}\left(-4x\right)+\frac{d}{dx}\left(-5\right)$

$12x^{2}+18x+\frac{d}{dx}\left(-4x\right)+0$

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

$12x^{2}+18x+\frac{d}{dx}\left(-4x\right)$
9

The derivative of the constant function ($-5$) is equal to zero

$12x^{2}+18x+\frac{d}{dx}\left(-4x\right)$

The derivative of a function multiplied by a constant ($-4$) is equal to the constant times the derivative of the function

$-4\frac{d}{dx}\left(x\right)$

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

$-4$
10

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

$12x^{2}+18x-4$

$12x^{2}+18x-4$