Derivative Rules Summary
Constant Rule
The derivative of any constant is zero. If f(x)=c, then f'(x)=0.
f(x)=15
f'(x)=0
g(x)=−π
g'(x)=0
h(x)=e^2
h'(x)=0
y=7
y'=0
s(t)=1/2
s'(t)=0
f(x)=sin(6π)
f'(x)=0
Constant Multiple Rule
The derivative of a constant times a function is the constant times the derivative of the function. If
f(x)=c⋅g(x), then f'(x)=c⋅g'(x).
f(x)=5x^3
f'(x)=15x^2
g(x)=(1/2)sin(x)
g'(x)=(1/2)cos(x)
h(x)=−4e^x
h'(x)=−4e^x
y=8ln(x)
y'=8/x
f(x)=(3/2)x^(1/2) f'(x)=(3/2)(1/2)x^(−1/2)=3/(2√x)
y=10x^(−2)
y'=−20/x^3
Sum Rule
The derivative of a sum of functions is the sum of their derivatives. If f(x)=g(x)+h(x), then
f'(x)=g'(x)+h'(x).
f(x)=x^2 + 4x
f'(x)=2x+4
g(x)=sin(x)+e^x
g'(x)=cos(x)+e^x
h(x)=7x^5 + ln(x)
h'(x)=35x^4 + 1/x
y=x^4 + x^(−1) + 3
f(x)=5√x + 1/x
f(t)=t^3 + cos(t)−9
y'=4x^3 − 1/x^2
f'(x)=5/(2√x) − 1/x^2
f'(t)=3t^2 − sin(t)
Difference Rule
The derivative of a difference of functions is the difference of their derivatives. If f(x)=g(x)−h(x), then
f'(x)=g'(x)−h'(x).
f(x)=3x^4 − 2x^2
f'(x)=12x^3 − 4x
g(x)=cos(x)−e^x
g'(x)=−sin(x)−e^x
h(x)=ln(x)−5x
h'(x)=1/x − 5
Derivative Rules Summary
y=12x−sin(x)
y'=12−cos(x)
f(t)=t^(−3) − 4t^2
f'(t)=−3t^(−4) − 8t
y=x^(1/3) − x^(2/3)
y'=1/(3x^(2/3)) − 2/(3x^(1/3))
Product Rule
If f(x)=g(x)⋅h(x), then f'(x)=g'(x)h(x)+h'(x)g(x).
f(x)=x^2 sin(x)
f'(x)=x^2 cos(x)+2xsin(x)
g(x)=e^x cos(x)
g'(x)=e^x cos(x)−e^x sin(x)
h(x)=(x^3 − 2x)(ln(x))
y=(x+1)(x^2 +3x−2)
f(t)=√t e^t
f(x)=3x^2 tan(x)
h'(x)=(x^3 − 2x)(1/x)+(ln(x))(3x^2 − 2)
y'=3x^2 + 8x + 1
f'(t)=√t e^t + (1/(2√t)) e^t
f'(x)=3x^2 sec^2(x) + 6x tan(x)
Quotient Rule
If f(x)=h(x)/g(x), then f'(x)= [g(x)h'(x)−h(x)g'(x)] / [g(x)]^2
f(x)=(x−1)/(x+1)
f'(x)=−2/(x+1)^2
g(x)=cos(x)/sin(x)
g'(x)=sec^2(x)
h(x)=e^x/x^2
h'(x)=e^x(2x−x^2)/x^4
y=x/ln(x)
f(x)=(x^3+1)/e^x
y=(x^2+1)/1
y'= [1−ln(x)] / [ln(x)]^2
f'(x)=e^x(x^3−3x^2+1)/e^(2x)
y'=0
Chain Rule
The derivative of a composite function f(g(x)) is f'(g(x))⋅g'(x).
f(x)=(x^2 +1)^3
f'(x)=6x(x^2 +1)^2
g(x)=sin(4x)
g'(x)=4cos(4x)
h(x)=e^(5x)
h'(x)=5e^(5x)
y=ln(x^3 + x)
y'=(3x^2 + 1)/(x^3 + x)
f(x)=√(x^2 −3)
f'(x)=x/√(x^2 −3)
y=cos^2(x)
y'=−2sin(x)cos(x)