From Wikipedia, the free encyclopedia
(Click for larger image) At each point, the derivative of
f
(
x
)
=
1
+
x
sin
x
2
{\displaystyle \scriptstyle f(x)=1+x\sin x^{2}}
slope of a line that is tangent to the curve . The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where green, negative where red, and zero where black.
m
=
change in
y
change in
x
=
Δ
y
Δ
x
{\displaystyle m={{\mbox{change in }}y \over {\mbox{change in }}x}={\Delta y \over {\Delta x}}}
d
y
d
x
{\displaystyle {\frac {dy}{dx}}\,\!}
m
=
Δ
f
(
x
)
Δ
x
=
f
(
x
+
h
)
−
f
(
x
)
h
.
{\displaystyle m={\frac {\Delta f(x)}{\Delta x}}={\frac {f(x+h)-f(x)}{h}}.}
f
′
(
a
)
=
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
{\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
−
f
′
(
a
)
⋅
h
h
=
0
,
{\displaystyle \lim _{h\to 0}{f(a+h)-f(a)-f'(a)\cdot h \over h}=0,}
f
(
a
+
h
)
≈
f
(
a
)
+
f
′
(
a
)
h
{\displaystyle f(a+h)\approx f(a)+f'(a)h}
Q
(
h
)
=
f
(
a
+
h
)
−
f
(
a
)
h
.
{\displaystyle Q(h)={\frac {f(a+h)-f(a)}{h}}.}
Q (h ) is the slope of the secant line between (a , ç'( a)) and ( a + h, ç'(a + h )). If ç' is a continuous function , meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from the point h = 0. If the limit
lim
h
→
0
Q
(
h
)
{\displaystyle \textstyle \lim _{h\to 0}Q(h)}
Q(0) that makes the graph of Q a continuous function, then the function ç' is differentiable at the point a , and its derivative at a equals Q (0).
f
′
(
3
)
=
lim
h
→
0
f
(
3
+
h
)
−
f
(
3
)
h
=
lim
h
→
0
(
3
+
h
)
2
−
9
h
=
lim
h
→
0
9
+
6
h
+
h
2
−
9
h
=
lim
h
→
0
6
h
+
h
2
h
=
lim
h
→
0
6
+
h
.
{\displaystyle f'(3)=\lim _{h\to 0}{f(3+h)-f(3) \over h}=\lim _{h\to 0}{(3+h)^{2}-9 \over {h}}=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}=\lim _{h\to 0}{6h+h^{2} \over {h}}=\lim _{h\to 0}{6+h}.}
lim
h
→
0
6
+
h
=
6
+
0
=
6.
{\displaystyle \lim _{h\to 0}{6+h}=6+0=6.}
1
↦
2
,
2
↦
4
,
3
↦
6.
{\displaystyle {\begin{aligned}1&{}\mapsto 2,\\2&{}\mapsto 4,\\3&{}\mapsto 6.\end{aligned}}}
D
(
x
↦
1
)
=
(
x
↦
0
)
,
D
(
x
↦
x
)
=
(
x
↦
1
)
,
D
(
x
↦
x
2
)
=
(
x
↦
2
⋅
x
)
.
{\displaystyle {\begin{aligned}D(x\mapsto 1)&=(x\mapsto 0),\\D(x\mapsto x)&=(x\mapsto 1),\\D(x\mapsto x^{2})&=(x\mapsto 2\cdot x).\end{aligned}}}
x
↦
x
2
,
{\displaystyle x\mapsto x^{2},}
x
↦
2
x
,
{\displaystyle x\mapsto 2x,}
f
(
x
)
=
{
x
2
,
if
x
≥
0
−
x
2
,
if
x
≤
0.
{\displaystyle f(x)={\begin{cases}x^{2},&{\mbox{if }}x\geq 0\\-x^{2},&{\mbox{if }}x\leq 0.\end{cases}}}
f
′
(
x
)
=
{
2
x
,
if
x
≥
0
−
2
x
,
if
x
≤
0.
{\displaystyle f'(x)={\begin{cases}2x,&{\mbox{if }}x\geq 0\\-2x,&{\mbox{if }}x\leq 0.\end{cases}}}
f
(
x
+
h
)
≈
f
(
x
)
+
f
′
(
x
)
h
+
1
2
f
″
(
x
)
h
2
{\displaystyle f(x+h)\approx f(x)+f'(x)h+{\tfrac {1}{2}}f''(x)h^{2}}
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
−
f
′
(
x
)
h
−
1
2
f
″
(
x
)
h
2
h
2
=
0.
{\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)-f'(x)h-{\frac {1}{2}}f''(x)h^{2}}{h^{2}}}=0.}
d
y
d
x
,
d
f
d
x
(
x
)
,
o
r
d
d
x
f
(
x
)
,
{\displaystyle {\frac {dy}{dx}},\quad {\frac {df}{dx}}(x),\;\;\mathrm {or} \;\;{\frac {d}{dx}}f(x),}
d
n
y
d
x
n
,
d
n
f
d
x
n
(
x
)
,
o
r
d
n
d
x
n
f
(
x
)
{\displaystyle {\frac {d^{n}y}{dx^{n}}},\quad {\frac {d^{n}f}{dx^{n}}}(x),\;\;\mathrm {or} \;\;{\frac {d^{n}}{dx^{n}}}f(x)}
d
2
y
d
x
2
=
d
d
x
(
d
y
d
x
)
.
{\displaystyle {\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}\left({\frac {dy}{dx}}\right).}
d
y
d
x
|
x
=
a
=
d
y
d
x
(
a
)
.
{\displaystyle \left.{\frac {dy}{dx}}\right|_{x=a}={\frac {dy}{dx}}(a).}
d
y
d
x
=
d
y
d
u
⋅
d
u
d
x
.
{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}
(
f
′
)
′
=
f
″
{\displaystyle (f')'=f''\,}
(
f
″
)
′
=
f
‴
.
{\displaystyle (f'')'=f'''.\,}
f
i
v
{\displaystyle f^{\mathrm {iv} }\,\!}
f
(
4
)
.
{\displaystyle f^{(4)}.\,\!}
y
˙
{\displaystyle {\dot {y}}}
y
¨
{\displaystyle {\ddot {y}}}
D
x
y
{\displaystyle D_{x}y\,}
D
x
f
(
x
)
{\displaystyle D_{x}f(x)\,}
f
(
x
)
=
x
r
,
{\displaystyle f(x)=x^{r},\,}
f
′
(
x
)
=
r
x
r
−
1
,
{\displaystyle f'(x)=rx^{r-1},\,}
wherever this function is defined. For example, if
f
(
x
)
=
x
1
/
4
{\displaystyle f(x)=x^{1/4}}
f
′
(
x
)
=
(
1
/
4
)
x
−
3
/
4
,
{\displaystyle f'(x)=(1/4)x^{-3/4},\,}
d
d
x
e
x
=
e
x
{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}
d
d
x
a
x
=
ln
(
a
)
a
x
{\displaystyle {\frac {d}{dx}}a^{x}=\ln(a)a^{x}}
d
d
x
ln
(
x
)
=
1
x
,
x
>
0
{\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}},\qquad x>0}
d
d
x
log
a
(
x
)
=
1
x
ln
(
a
)
{\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}}}
d
d
x
sin
(
x
)
=
cos
(
x
)
.
{\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x).}
d
d
x
cos
(
x
)
=
−
sin
(
x
)
.
{\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x).}
d
d
x
tan
(
x
)
=
sec
2
(
x
)
=
1
cos
2
(
x
)
=
1
+
tan
2
(
x
)
.
{\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x).}
d
d
x
arcsin
(
x
)
=
1
1
−
x
2
.
{\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}.}
d
d
x
arccos
(
x
)
=
−
1
1
−
x
2
.
{\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}.}
d
d
x
arctan
(
x
)
=
1
1
+
x
2
.
{\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}.}
f
′
=
0
{\displaystyle f'=0\,}
(
a
f
+
b
g
)
′
=
a
f
′
+
b
g
′
{\displaystyle (af+bg)'=af'+bg'\,}
ƒ and g and all real numbers a and b .
(
f
g
)
′
=
f
′
g
+
f
g
′
{\displaystyle (fg)'=f'g+fg'\,}
ƒ and g .
(
f
g
)
′
=
f
′
g
−
f
g
′
g
2
{\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-fg'}{g^{2}}}}
ƒ and g where g ? 0.Chain rule
f
(
x
)
=
h
(
g
(
x
)
)
{\displaystyle f(x)=h(g(x))}
f
′
(
x
)
=
h
′
(
g
(
x
)
)
⋅
g
′
(
x
)
.
{\displaystyle f'(x)=h'(g(x))\cdot g'(x).\,}
f
(
x
)
=
x
4
+
sin
(
x
2
)
−
ln
(
x
)
e
x
+
7
{\displaystyle f(x)=x^{4}+\sin(x^{2})-\ln(x)e^{x}+7\,}
f
′
(
x
)
=
4
x
(
4
−
1
)
+
d
(
x
2
)
d
x
cos
(
x
2
)
−
d
(
ln
x
)
d
x
e
x
−
ln
x
d
(
e
x
)
d
x
+
0
=
4
x
3
+
2
x
cos
(
x
2
)
−
1
x
e
x
−
ln
(
x
)
e
x
.
{\displaystyle {\begin{aligned}f'(x)&=4x^{(4-1)}+{\frac {d\left(x^{2}\right)}{dx}}\cos(x^{2})-{\frac {d\left(\ln {x}\right)}{dx}}e^{x}-\ln {x}{\frac {d\left(e^{x}\right)}{dx}}+0\\&=4x^{3}+2x\cos(x^{2})-{\frac {1}{x}}e^{x}-\ln(x)e^{x}.\end{aligned}}}
y
′
(
t
)
=
(
y
1
′
(
t
)
,
…
,
y
n
′
(
t
)
)
.
{\displaystyle \mathbf {y} '(t)=(y'_{1}(t),\ldots ,y'_{n}(t)).}
y
′
(
t
)
=
lim
h
→
0
y
(
t
+
h
)
−
y
(
t
)
h
,
{\displaystyle \mathbf {y} '(t)=\lim _{h\to 0}{\frac {\mathbf {y} (t+h)-\mathbf {y} (t)}{h}},}
y
1
′
(
t
)
e
1
+
⋯
+
y
n
′
(
t
)
e
n
{\displaystyle y'_{1}(t)\mathbf {e} _{1}+\cdots +y'_{n}(t)\mathbf {e} _{n}}
f
(
x
,
y
)
=
x
2
+
x
y
+
y
2
.
{\displaystyle f(x,y)=x^{2}+xy+y^{2}.\,}
f
(
x
,
y
)
=
f
x
(
y
)
=
x
2
+
x
y
+
y
2
.
{\displaystyle f(x,y)=f_{x}(y)=x^{2}+xy+y^{2}.\,}
x
↦
f
x
,
{\displaystyle x\mapsto f_{x},\,}
f
x
(
y
)
=
x
2
+
x
y
+
y
2
.
{\displaystyle f_{x}(y)=x^{2}+xy+y^{2}.\,}
f
a
(
y
)
=
a
2
+
a
y
+
y
2
.
{\displaystyle f_{a}(y)=a^{2}+ay+y^{2}.\,}
f
a
′
(
y
)
=
a
+
2
y
.
{\displaystyle f_{a}'(y)=a+2y.\,}
∂
f
∂
y
(
x
,
y
)
=
x
+
2
y
.
{\displaystyle {\frac {\partial f}{\partial y}}(x,y)=x+2y.}
∂
f
∂
x
i
(
a
1
,
…
,
a
n
)
=
lim
h
→
0
f
(
a
1
,
…
,
a
i
+
h
,
…
,
a
n
)
−
f
(
a
1
,
…
,
a
n
)
h
.
{\displaystyle {\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i}+h,\ldots ,a_{n})-f(a_{1},\ldots ,a_{n})}{h}}.}
f
a
1
,
…
,
a
i
−
1
,
a
i
+
1
,
…
,
a
n
(
x
i
)
=
f
(
a
1
,
…
,
a
i
−
1
,
x
i
,
a
i
+
1
,
…
,
a
n
)
{\displaystyle f_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}(x_{i})=f(a_{1},\ldots ,a_{i-1},x_{i},a_{i+1},\ldots ,a_{n})}
d
f
a
1
,
…
,
a
i
−
1
,
a
i
+
1
,
…
,
a
n
d
x
i
(
a
i
)
=
∂
f
∂
x
i
(
a
1
,
…
,
a
n
)
.
{\displaystyle {\frac {df_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}}{dx_{i}}}(a_{i})={\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n}).}
∇
f
(
a
)
=
(
∂
f
∂
x
1
(
a
)
,
…
,
∂
f
∂
x
n
(
a
)
)
.
{\displaystyle \nabla f(a)=\left({\frac {\partial f}{\partial x_{1}}}(a),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a)\right).}
v
=
(
v
1
,
…
,
v
n
)
.
{\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n}).}
D
v
f
(
x
)
=
lim
h
→
0
f
(
x
+
h
v
)
−
f
(
x
)
h
.
{\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\rightarrow 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.}
f
(
x
+
(
k
/
λ
)
(
λ
u
)
)
−
f
(
x
)
k
/
λ
=
λ
⋅
f
(
x
+
k
u
)
−
f
(
x
)
k
.
{\displaystyle {\frac {f(\mathbf {x} +(k/\lambda )(\lambda \mathbf {u} ))-f(\mathbf {x} )}{k/\lambda }}=\lambda \cdot {\frac {f(\mathbf {x} +k\mathbf {u} )-f(\mathbf {x} )}{k}}.}
D
v
f
(
x
)
=
∑
j
=
1
n
v
j
∂
f
∂
x
j
.
{\displaystyle D_{\mathbf {v} }{f}({\boldsymbol {x}})=\sum _{j=1}^{n}v_{j}{\frac {\partial f}{\partial x_{j}}}.}
f
(
a
+
v
)
≈
f
(
a
)
+
f
′
(
a
)
v
.
{\displaystyle f(\mathbf {a} +\mathbf {v} )\approx f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {v} .}
f
(
a
+
v
)
−
f
(
a
)
≈
f
′
(
a
)
v
.
{\displaystyle f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} )\approx f'(\mathbf {a} )\mathbf {v} .}
f
(
a
+
v
+
w
)
−
f
(
a
+
v
)
−
f
(
a
+
w
)
+
f
(
a
)
≈
f
′
(
a
+
v
)
w
−
f
′
(
a
)
w
.
{\displaystyle f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} +\mathbf {w} )+f(\mathbf {a} )\approx f'(\mathbf {a} +\mathbf {v} )\mathbf {w} -f'(\mathbf {a} )\mathbf {w} .}
0
≈
f
(
a
+
v
+
w
)
−
f
(
a
+
v
)
−
f
(
a
+
w
)
+
f
(
a
)
=
(
f
(
a
+
v
+
w
)
−
f
(
a
)
)
−
(
f
(
a
+
v
)
−
f
(
a
)
)
−
(
f
(
a
+
w
)
−
f
(
a
)
)
≈
f
′
(
a
)
(
v
+
w
)
−
f
′
(
a
)
v
−
f
′
(
a
)
w
.
{\displaystyle {\begin{aligned}0&\approx f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} +\mathbf {w} )+f(\mathbf {a} )\\&=(f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} ))-(f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} ))-(f(\mathbf {a} +\mathbf {w} )-f(\mathbf {a} ))\\&\approx f'(\mathbf {a} )(\mathbf {v} +\mathbf {w} )-f'(\mathbf {a} )\mathbf {v} -f'(\mathbf {a} )\mathbf {w} .\end{aligned}}}
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
−
f
′
(
a
)
h
h
=
0.
{\displaystyle \lim _{h\to 0}{\frac {f(a+h)-f(a)-f'(a)h}{h}}=0.}
lim
h
→
0
|
f
(
a
+
h
)
−
f
(
a
)
−
f
′
(
a
)
h
|
|
h
|
=
0
{\displaystyle \lim _{h\to 0}{\frac {|f(a+h)-f(a)-f'(a)h|}{|h|}}=0}
lim
‖
h
‖
→
0
‖
f
(
a
+
h
)
−
f
(
a
)
−
f
′
(
a
)
h
‖
‖
h
‖
=
0.
{\displaystyle \lim _{\lVert \mathbf {h} \rVert \to 0}{\frac {\lVert f(\mathbf {a} +\mathbf {h} )-f(\mathbf {a} )-f'(\mathbf {a} )\mathbf {h} \rVert }{\lVert \mathbf {h} \rVert }}=0.}
f
′
(
a
)
=
Jac
a
=
(
∂
f
i
∂
x
j
)
i
j
.
{\displaystyle f'(\mathbf {a} )=\operatorname {Jac} _{\mathbf {a} }=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{ij}.}
f
(
a
+
h
)
≈
f
(
a
)
+
f
′
(
a
)
h
.
{\displaystyle f(a+h)\approx f(a)+f'(a)h.}
Up to changing variables, this is the statement that the function
x
↦
f
(
a
)
+
f
′
(
a
)
(
x
−
a
)
{\displaystyle x\mapsto f(a)+f'(a)(x-a)}
ç' at a.
