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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

FunctionDerivativeIntegralGraph
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

FunctionDerivativeIntegralGraph
sinh(x)cosh(x)cosh(x)  ↓  
cosh(x)sinh(x)sinh(x)  ↓  
tanh(x)1-tanh(x)²ln(cosh(x))  ↓  
coth(x)1-coth(x)²ln(|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|<1x*artanh(x) + ln(1-x²)/2  ↓  
arcoth(x)-1/(x²-1); |x|>1x*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 )  ↓  


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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 - graphs
Sine: function, derivative and integral

Cosine: function, derivative and integral - graphs
Cosine: function, derivative and integral

Tangent: function, derivative and integral - graphs
Tangent: function, derivative and integral

Cotangent: function, derivative and integral - graphs
Cotangent: function, derivative and integral

Secant: function, derivative and integral - graphs
Secant: function, derivative and integral

Cosecant: function, derivative and integral - graphs
Cosecant: function, derivative and integral

Sine square: function, derivative and integral - graphs
Sine square: function, derivative and integral

Cosine square: function, derivative and integral - graphs
Cosine square: function, derivative and integral

Tangent square: function, derivative and integral - graphs
Tangent square: function, derivative and integral

Cotangent square: function, derivative and integral - graphs
Cotangent square: function, derivative and integral

Secant square: function, derivative and integral - graphs
Secant square: function, derivative and integral

Cosecant square: function, derivative and integral - graphs
Cosecant square: function, derivative and integral

Arcsine: function, derivative and integral - graphs
Arcsine: function, derivative and integral

Arccosine: function, derivative and integral - graphs
Arccosine: function, derivative and integral

Arctangent: function, derivative and integral - graphs
Arctangent: function, derivative and integral

Arccotangent: function, derivative and integral - graphs
Arccotangent: function, derivative and integral

Arcsecant: function, derivative and integral - graphs
Arcsecant: function, derivative and integral

Arccosecant: function, derivative and integral - graphs
Arccosecant: function, derivative and integral

Hyperbolic Functions

Hyperbolic sine: function, derivative and integral - graphs
Hyperbolic sine : function, derivative and integral

Hyperbolic cosine: function, derivative and integral - graphs
Hyperbolic cosine : function, derivative and integral

Hyperbolic tangent: function, derivative and integral - graphs
Hyperbolic tangent : function, derivative and integral

Hyperbolic cotangent: function, derivative and integral - graphs
Hyperbolic cotangent : function, derivative and integral

Hyperbolic secant: function, derivative and integral - graphs
Hyperbolic secant : function, derivative and integral

Hyperbolic cosecant: function, derivative and integral - graphs
Hyperbolic cosecant : function, derivative and integral

Area hyperbolic sine: function, derivative and integral - graphs
Area hyperbolic sine : function, derivative and integral

Area hyperbolic cosine: function, derivative and integral - graphs
Area hyperbolic cosine : function, derivative and integral

Area hyperbolic tangent: function, derivative and integral - graphs
Area hyperbolic tangent : function, derivative and integral

Area hyperbolic cotangent: function, derivative and integral - graphs
Area hyperbolic cotangent : function, derivative and integral

Area hyperbolic secant: function, derivative and integral - graphs
Area hyperbolic secant : function, derivative and integral

Area hyperbolic cosecant: function, derivative and integral - graphs
Area hyperbolic cosecant : function, derivative and integral


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



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The function graphs were made with the Function Graph Plotter.