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Sum Rule of Differentiation Calculator

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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 or more functions is the sum of the derivatives of each function

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

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

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

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

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

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

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

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

$-4\cdot 1$

Any expression multiplied by $1$ is equal to itself

$-4$
5

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

$\frac{d}{dx}\left(4x^3\right)+\frac{d}{dx}\left(9x^2\right)-4$
6

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

$4\frac{d}{dx}\left(x^3\right)+\frac{d}{dx}\left(9x^2\right)-4$
7

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

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

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

$12x^{\left(3-1\right)}$

Subtract the values $3$ and $-1$

$12x^{2}$
8

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

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

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

$18x^{\left(2-1\right)}$

Subtract the values $2$ and $-1$

$18x$
9

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

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

Final Answer

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

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