The definition of a Derivative of a Function
f
′
(
x
)
=
lim
Δ
x
→
0
f
(
x
+
Δ
x
)
−
f
(
x
)
Δ
x
{\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {f(x+{\Delta }x)-f(x)}{{\Delta }x}}}
Example
f
(
x
)
=
x
2
{\displaystyle f(x)=x^{2}}
f
′
(
x
)
=
lim
Δ
x
→
0
(
x
+
Δ
x
)
2
−
x
2
Δ
x
{\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {(x+{\Delta }x)^{2}-x^{2}}{{\Delta }x}}}
f
′
(
x
)
=
lim
Δ
x
→
0
(
x
2
+
2
x
Δ
x
+
Δ
x
2
)
−
x
2
Δ
x
{\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {(x^{2}+2x{\Delta }x+{\Delta }x^{2})-x^{2}}{{\Delta }x}}}
f
′
(
x
)
=
lim
Δ
x
→
0
2
x
Δ
x
+
Δ
x
2
Δ
x
{\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {2x{\Delta }x+{\Delta }x^{2}}{{\Delta }x}}}
f
′
(
x
)
=
lim
Δ
x
→
0
Δ
x
(
2
x
+
Δ
x
)
Δ
x
{\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}{\frac {{\Delta }x(2x+{\Delta }x)}{{\Delta }x}}}
f
′
(
x
)
=
lim
Δ
x
→
0
(
2
x
+
Δ
x
)
{\displaystyle f'(x)=\lim _{{\Delta }x\rightarrow 0}(2x+{\Delta }x)}
f
′
(
x
)
=
2
x
+
0
{\displaystyle f'(x)=2x+0}
f
′
(
x
)
=
2
x
{\displaystyle f'(x)=2x}
