Skip to content

Calculus

Derivatives

Derivative definition, notation, properties, common derivatives, higher-order derivatives, and critical points.

The derivative measures the instantaneous rate of change of a function. Geometrically it is the slope of the tangent line at a point; analytically it is the limit of a difference quotient. The rules below (constant multiple, sum, product, quotient, chain, power) reduce most differentiation to mechanical pattern application. The table of common derivatives gives the leaves; the rules combine them. The sign of f’ locates intervals of increase and decrease. The sign of f” distinguishes concave-up from concave-down behaviour and locates inflection points.

a(a, f(a))slope = f’(a)xy
f'(a) is the slope of the tangent line to f at x = a.

Definition

The derivative is the limit of a difference quotient, the slope of the secant as the two points coalesce.

If y = f(x), the derivative is

f′(x)=lim⁡h→c(f(x+h)−f(x)h)f'\left(x\right)=\lim_{h \rightarrow c}\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)

Notations for the derivative are interchangeable: Lagrange’s f’, Leibniz’s dy/dx, operator D.

f′(x)=y′=dfdx=dydx=ddx(f(x))=Df(x)f'(x)=y'=\frac{df}{dx}=\frac{dy}{dx}=\frac{d}{dx}\left(f\left(x\right)\right)=Df\left(x\right)

At a point x = a, the derivative is written with a vertical bar or as f’(a):

f′(a)=y′∣x=a=dfdx∣x=a=dydx∣x=a=Df(a)f'\left(a\right)=\left.y'\right|_{x=a}=\left.\frac{df}{dx}\right|_{x=a}=\left.\frac{dy}{dx}\right|_{x=a}=Df\left(a\right)

Interpretation of Derivatives

Three standard interpretations: geometric (tangent slope), analytic (instantaneous rate of change), and physical (velocity).

if y = f(x) then,

  1. m = f’(a) is the slope of the tangent line to y = f(x) at x = a and the equation of the tangent line at x = a is given by y = f(a)+f’(a)(x-a).

  2. f’(a) is the instantaneous rate of change of f(x) at x = a.

  3. If f(x) is the position of an object at time x then f’(a) is the velocity of the object at x = a.

Basic Properties

Rules for sums, products, quotients, powers, and compositions.

Constants pass through:

(cf)′=cf′(x)\left(cf\right)'=cf'\left(x\right)

Sum and difference distribute:

(f±g)′=f′(x)±g′(x)\left(f\pm g\right)'=f'\left(x\right)\pm g'\left(x\right)

Product Rule, symmetric in f and g:

(fg)′=f′g+fg′    Product Rule\left(fg\right)'=f'g+fg'\text{ }\text{ }\text{ }\text{ }Product\text{ }Rule

Quotient Rule, g ≠ 0:

(fg)′=(f′g−fg′g2)    Quotient Rule\left(\frac{f}{g}\right)'=\left(\frac{f'g-fg'}{g^{2}}\right)\text{ }\text{ }\text{ }\text{ }Quotient\text{ }Rule

A constant has derivative zero:

ddx(c)=0\frac{d}{dx}\left(c\right)=0

Power Rule, any real exponent:

ddx(xn)=nxn−1    Power Rule\frac{d}{dx}\left(x^{n}\right)=nx^{n-1}\text{ }\text{ }\text{ }\text{ }Power\text{ }Rule

Chain Rule: outer derivative (evaluated at the inner function) times the inner derivative. Used whenever one function appears inside another.

ddx(f(g(x)))=f′(g(x))g′(x)     Chain Rule\frac{d}{dx}\left(f\left(g\left(x\right)\right)\right)=f'\left(g\left(x\right)\right)g'\left(x\right)\text{ }\text{ }\text{ }\text{ }\text{ }Chain\text{ }Rule

Common Derivatives

The elementary derivatives. Combined with the rules above they handle most closed-form differentiation.

Identity:

ddx(x)=1\frac{d}{dx}\left(x\right)=1

Sine:

ddx(sin⁡x)=cos⁡x\frac{d}{dx}\left(\sin x\right)=\cos x

Cosine:

ddx(cos⁡x)=−sin⁡x\frac{d}{dx}\left(\cos x\right)=-\sin x

Tangent:

ddx(tan⁡x)=sec⁡2x\frac{d}{dx}\left(\tan x\right)=\sec^{2} x

Secant:

ddx(sec⁡x)=sec⁡xtan⁡x\frac{d}{dx}\left(\sec x\right)=\sec x\tan x

Cosecant:

ddx(csc⁡x)=csc⁡xcot⁡x\frac{d}{dx}\left(\csc x\right)=\csc x\cot x

Cotangent:

ddx(cot⁡x)=−csc⁡2x\frac{d}{dx}\left(\cot x\right)=-\csc^{2} x

Arcsine:

ddx(sin⁡−1x)=11−x2\frac{d}{dx}\left(\sin^{-1} x\right)=\frac{1}{\sqrt{1-x^{2}}}

Arccosine:

ddx(cos⁡−1x)=−11−x2\frac{d}{dx}\left(\cos^{-1} x\right)=-\frac{1}{\sqrt{1-x^{2}}}

Arctangent:

ddx(tan⁡−1x)=11+x2\frac{d}{dx}\left(\tan^{-1} x\right)=\frac{1}{{1+x^{2}}}

Exponential, base a:

ddx(ax)=axln⁡(a)\frac{d}{dx}\left(a^{x}\right)=a^{x}\ln\left(a\right)

Natural exponential, its own derivative:

ddx(ex)=ex\frac{d}{dx}\left(e^{x}\right)=e^{x}

Natural log, x > 0:

ddx(ln⁡(x))=1x    ,x>0\frac{d}{dx}\left(\ln\left(x\right)\right)=\frac{1}{x}\text{ }\text{ }\text{ }\text{ },x>0

Extended to ln|x| for x ≠ 0:

ddx(ln⁡∣x∣)=1x    ,x≠0\frac{d}{dx}\left(\ln\left|x\right|\right)=\frac{1}{x}\text{ }\text{ }\text{ }\text{ },x\neq0

Log, base a:

ddx(log⁡a(x))=1xln⁡a    ,x>0\frac{d}{dx}\left(\log_{a}{\left(x\right)}\right)=\frac{1}{x\ln{a}}\text{ }\text{ }\text{ }\text{ },x>0

Higher Order Derivatives

Repeated differentiation produces second, third, and higher derivatives.

The second derivative:

 f′′(x)=f(2)(x)=d2fdx2 \text{ }f''\left(x\right)=f^{\left(2\right)}\left(x\right)=\frac{d^{2}f}{dx^{2}}\text{ }

is the derivative of the first derivative:

 f′′(x)=(f′(x))′ \text{ }f''\left(x\right)=\left(f'\left(x\right)\right)'\text{ }

The n-th derivative:

 f(n)(x)=dnfdxn \text{ }f^{\left(n\right)}\left(x\right)=\frac{d^{n}f}{dx^{n}}\text{ }

is defined recursively from the (n-1)-th derivative:

 f(n)(x)=(f(n−1)(x))′ \text{ }f^{\left(n\right)}\left(x\right)=\left(f^{\left(n-1\right)}\left(x\right)\right)'\text{ }

, i.e. the derivative of the (n-1)st derivative, f(n-1)(x).

Others

The first and second derivatives classify behaviour: critical points (where f’ is zero or undefined), monotonicity (sign of f’), concavity (sign of f”).

Critical Points

x = c is a critical point of f(x) provied either

  1. f’(c) = 0

  2. f’(c) doesn’t exist.

Increasing/Decreasing

  1. If f’(x) > 0 for all x in an interval I then f (x) is increasing on the interval I.

  2. If f’(x) < 0 for all x in an interval I then f (x) is Decreasing on the interval I.

  3. If f’(x) = 0 for all x in an interval I then f (x) is constant on the interval I.

Concave up/Concave down

  1. If f”(x) > 0 for all x in an interval I then f (x) is concave up on the interval I.

  2. If f”(x) < 0 for all x in an interval I then f (x) is concave down on the interval I.

Infection Points

x = c is inflection point of f(x) if the concavity changes at x = c.

f”(x) > 0f”(x) < 0concave upconcave down
f''(x) > 0 means concave up; f''(x) < 0 means concave down.