Derivative and Integral of Trigonometric and Hyperbolic Functions

Formulas and graphs of derivatives and integrals of the trigonometric and hyperbolic functions. For the formulas of the integrals, the +C is omitted. Clicking ↓ shows the according graph.

Formulas

Trigonometric Functions

Function	Derivative	Integral	Graph
sin(x)	cos(x)	-cos(x)	↓
cos(x)	-sin(x)	sin(x)	↓
tan(x)	1+tan²(x)	-ln(|cos(x)|)	↓
cot(x)	-1-cot²(x)	ln(|sin(x)|)	↓
sec(x)	sec²(x)/csc(x)	ln(|sec(x)+tan(x)|)	↓
csc(x)	-csc²(x)/sec(x)	ln(|tan(x/2)|)	↓
sin²(x)	sin(2x)	(x-sin(x)*cos(x))/2	↓
cos²(x)	-2*sin(x)*cos(x)	(x+sin(x)*cos(x))/2	↓
tan²(x)	2*sec²(x)*tan(x)	tan(x)-x	↓
cot²(x)	-2csc²(x)*cot(x)	-cot(x)-x	↓
sec²(x)	2*sec²(x)*tan(x)	tan(x)	↓
csc²(x)	-2*csc²(x)*cot(x)	-cot(x)	↓
asin(x)	1/√ 1-x²	x*asin(x) + √ 1-x²	↓
acos(x)	-1/√ 1-x²	x*acos(x) - √ 1-x²	↓
atan(x)	1/(1+x²)	x*atan(x) - ln(1+x²)/2	↓
acot(x)	-1/(1+x²)	x*acot(x) + ln(1+x²)/2	↓
asec(x)	1/(|x|*√ |x|-1 )	x*asec(x) - arcosh(|x|)	↓
acsc(x)	-1/(|x|*√ |x|-1 )	x*acsc(x) + arcosh(|x|)	↓

Hyperbolic Functions

Function	Derivative	Integral	Graph
sinh(x)	cosh(x)	cosh(x)	↓
cosh(x)	sinh(x)	sinh(x)	↓
tanh(x)	1-tanh(x)²	ln(cosh(x))	↓
coth(x)	1-coth(x)²	ln(|sinh(x)|)	↓
sech(x)	-sech(x)*tanh(x)	atan(sinh(x))	↓
csch(x)	-coth(x)*csch(x)	ln(|tanh(x/2)|)	↓
arsinh(x)	1/√ x²+1	x*arsinh(x) - √ x²+1	↓
arcosh(x)	1/√ x²-1	x*arcosh(x) - √ x²-1	↓
artanh(x)	1/(1-x²); |x|<1	x*artanh(x) + ln(1-x²)/2	↓
arcoth(x)	-1/(x²-1); |x|>1	x*arcoth(x) + ln(x²-1)/2	↓
arsech(x)	-1/(x*√ 1-x² )	x*arsech(x) - 2*atan(√ (1-x)/(1+x) )	↓
arcsch(x)	-1/(|x|*√ 1+x² )	x*arcsch(x) + arcoth(√ 1/x²+1 )	↓

Graphs

The graphs of function, derivative and integral of trigonometric and hyperbolic functions in one image each. The graph of a function f is blue, that one of the derivative g is red and that of an integral h is green. abs is the absolute value, sqr is the square root and ln is the natural logarithm.

Trigonometric Functions

Sine: function, derivative and integral


Cosine: function, derivative and integral


Tangent: function, derivative and integral


Cotangent: function, derivative and integral


Secant: function, derivative and integral


Cosecant: function, derivative and integral


Sine square: function, derivative and integral


Cosine square: function, derivative and integral


Tangent square: function, derivative and integral


Cotangent square: function, derivative and integral


Secant square: function, derivative and integral


Cosecant square: function, derivative and integral


Arcsine: function, derivative and integral


Arccosine: function, derivative and integral


Arctangent: function, derivative and integral


Arccotangent: function, derivative and integral


Arcsecant: function, derivative and integral


Arccosecant: function, derivative and integral

Hyperbolic Functions

Hyperbolic sine : function, derivative and integral


Hyperbolic cosine : function, derivative and integral


Hyperbolic tangent : function, derivative and integral


Hyperbolic cotangent : function, derivative and integral


Hyperbolic secant : function, derivative and integral


Hyperbolic cosecant : function, derivative and integral


Area hyperbolic sine : function, derivative and integral


Area hyperbolic cosine : function, derivative and integral


Area hyperbolic tangent : function, derivative and integral


Area hyperbolic cotangent : function, derivative and integral


Area hyperbolic secant : function, derivative and integral


Area hyperbolic cosecant : function, derivative and integral


This page in German: Ableitung und Stammfunktion von trigonometrischen Funktionen und Hyperbelfunktionen.


The function graphs were made with the Function Graph Plotter.