1. calculators
  2. Sum Rule Of Differentiation

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!

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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

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

$\frac{d}{dx}\left(4x^3\right)+\frac{d}{dx}\left(9x^2\right)+\frac{d}{dx}\left(-4x\right)+0$

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

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

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

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

$-4$
4

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

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

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

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

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

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

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

Subtract the values $3$ and $-1$

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

Multiply $4$ times $3$

$12x^{2}+9\frac{d}{dx}\left(x^2\right)-4$
7

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

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

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

Subtract the values $3$ and $-1$

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

Multiply $4$ times $3$

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

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

Subtract the values $2$ and $-1$

$12x^{2}+9\cdot 2x^{1}-4$

Multiply $9$ times $2$

$12x^{2}+18x^{1}-4$

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

$12x^{2}+18x-4$
8

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

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

Final Answer

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

Struggling with math?

Access detailed step by step solutions to thousands of problems, growing every day!