Formulas, Rules, & Notations

Definition of Derivative:

f'(x) = showing limit as h appoaches zeroopening bracket
f(x + h) - f(x)
h
closing bracket

Note: Assuming the limit exists and is finite.

Notations For The Derivative of a function f(x):

f '(x)

Df(x)

d over dx derivative operator f(x)

Tangent Line Equation:

The line y = f(a) + f '(a). ( x – a ) is tangent to the graph of f at (a, f(a) ) .

Derivative of a constant:

D( constant ) = 0

The Power Rule

D( xn ) = n.x n-1

Derivatives of Sine and Cosine

D( sin(x) ) = cos(x)

D( cos(x) ) = –sin(x)

Derivative of an Absolute Value

D( |x| ) = +1 if x > 0, undefined if x = 0, –1 if x < 0

Another Derivative Formula / Rule Table

Standard Function

The Function

Its Derivative

Constant

Any constant, c

0

Straight Line

x

1


ax, for a constant a

a

Square

x2

2x

Square Root

√(x)

(½)x(-½)

Exponential

ex

ex


ax

ln(a) (ax)

Logarithms

ln(x)

1
x

loga(x) 

1
x ln(a)
Trigonometry

cos(x)

-sin(x)


tan(x)

cos(x)


sin(x)

sec2(x)

Inverse Trigonometry

sin-1(x)

1
√(1-x2)

tan-1(x)

1
(1+x2)

cos-1(x)

1
√(1-x2)

Credit: https://www.mathsisfun.com/calculus/derivatives-rules.html