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Derivative rules

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Derivative rules

Definition
f′(x) = limh → 0 f(x + h) − f(x)h
The slope of the tangent line.Lesson: The derivative from the definition →
Power rule
ddx xn = nxn−1
Bring the power down, take 1 off the power.Lesson: The power rule →
Constant
ddx c = 0
Lesson: The power rule →
Constant multiple and sum
(cf)′ = cf′ (f + g)′ = f′ + g′
Lesson: The power rule →
Product rule
(fg)′ = f′g + fg′
Lesson: Product and quotient rules →
Quotient rule
(fg)′ = f′g − fg′g2
Lesson: Product and quotient rules →
Chain rule
ddx f(g(x)) = f′(g(x)) × g′(x)
Outside derivative × inside derivative.Lesson: The chain rule →

Derivatives of functions

sin and cos
ddx sin x = cos x ddx cos x = −sin x
Lesson: Derivatives of sin, cos, e and ln →
tan
ddx tan x = sec2 x
Lesson: Derivatives of sin, cos, e and ln →
eˣ and ln x
ddx ex = ex ddx ln x = 1x
Lesson: Derivatives of sin, cos, e and ln →
bˣ and log
ddx bx = bx ln b ddx logb x = 1x ln b
Lesson: Derivatives of bˣ / Derivatives of logarithms →
Inverse sine
ddx sin−1 x = 1√1 − x2
Lesson: Derivatives of inverse trig functions →
Inverse tangent
ddx tan−1 x = 11 + x2
Lesson: Derivatives of inverse trig functions →

Using derivatives

Implicit differentiation
ddx y2 = 2ydydx
Differentiate both sides; every y gets × dy/dx.Lesson: Implicit differentiation →
Tangent line
y − f(a) = f′(a)(x − a)
Lesson: Tangent lines →
Linear approximation
L(x) = f(a) + f′(a)(x − a)
Near a, the tangent line is close to f.Lesson: Linear approximation →
Differential
dy = f′(x) dx
Lesson: Differentials →
Critical points
f′(x) = 0 or undefined
Where a max or min can be.Lesson: Maximum, minimum and optimization →
Second derivative test
f″(c) > 0: min f″(c) < 0: max
At a point where f′(c) = 0.Lesson: The second derivative test →
Concavity
f″ > 0: up f″ < 0: down
Inflection point: where f″ changes sign.Lesson: Concavity and inflection points →
Mean value theorem
f′(c) = f(b) − f(a)b − a
Somewhere the slope equals the average slope.Lesson: The mean value theorem / Rolle's theorem →
Newton's method
xn+1 = xn − f(xn)f′(xn)
Lesson: Newton's method →
Logarithmic differentiation
ln y = ln f(x) → y′y = ddx ln f(x)
For powers like xˣ.Lesson: Logarithmic differentiation →

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