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# Find the derivative of $x^2$ using the definition

## Step-by-step Solution

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###  Videos

$2x$
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##  Step-by-step Solution 

Problem to solve:

$derivdef\left(x^2\right)$

Specify the solving method

1

Find the derivative of $x^2$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $x^2$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get

$\lim_{h\to0}\left(\frac{\left(x+h\right)^2-x^2}{h}\right)$

Square of the first term: $\left(x\right)^2 = [a^2]$.

Double product of the first by the second: $2\left(x\right)\left(h\right) = [2ab]$.

Square of the second term: $\left(h\right)^2 = [b^2]$.

2

Expand $\left(x+h\right)^2$

$\lim_{h\to0}\left(\frac{x^2+2xh+h^2-x^2}{h}\right)$
3

Cancel like terms $x^2$ and $-x^2$

$\lim_{h\to0}\left(\frac{2xh+h^2}{h}\right)$
4

Factor the polynomial $2xh+h^2$ by it's greatest common factor (GCF): $h$

$\lim_{h\to0}\left(\frac{h\left(2x+h\right)}{h}\right)$
5

Simplify the fraction $\frac{h\left(2x+h\right)}{h}$ by $h$

$\lim_{h\to0}\left(2x+h\right)$
6

Evaluate the limit $\lim_{h\to0}\left(2x+h\right)$ by replacing all occurrences of $h$ by $0$

$2x+0$
7

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

$2x$

$2x$

SnapXam A2

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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
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

### Main topic:

Definition of Derivative

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