Derivative Rules

The Derivative tells us the slope of a function at any point.

There are rules we can follow to find many derivatives.

For example:

• The slope of a constant value (like 3) is always 0
• The slope of a line like 2x is 2, or 3x is 3 etc
• and so on.

Here are useful rules to help you work out the derivatives of many functions (with examples below). Note: the little mark ’ means "Derivative of", and f and g are functions.

Common Functions	Function	Derivative
Constant	c	0
Line	x	1
	ax	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 (x is in radians)	sin(x)	cos(x)
	cos(x)	−sin(x)
	tan(x)	sec2(x)
Inverse Trigonometry	sin-1(x)	1/√(1−x2)
	cos-1(x)	−1/√(1−x2)
	tan-1(x)	1/(1+x2)
Rules	Function	Derivative
Multiplication by constant	cf	cf’
Power Rule	xn	nxn−1
Sum Rule	f + g	f’ + g’
Difference Rule	f - g	f’ − g’
Product Rule	fg	f g’ + f’ g
Quotient Rule	f/g	(f’ g − g’ f )/g2
Reciprocal Rule	1/f	−f’/f2
Chain Rule (as "Composition of Functions")	f º g	(f’ º g) × g’
Chain Rule (using ’ )	f(g(x))	f’(g(x))g’(x)
Chain Rule (using ddx )	dydx = dydududx