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 ) | ↓ |
Derivative and Integral
The derivative of a function is the gradient at each point of that function. There are numerous derivative rules that make differentiation much easier. The original method of differential calculus using differential quotients, on the other hand, is considerably more complicated. A well-known derivative rule states that the derivative of x to the power of n is equal to n times x to the power of n-1. Therefore, the derivative of the squared sine of x is equal to the sine of 2x. The derivative rules for trigonometric functions are usually more complicated, but the derivative of sine is cosine, and the derivative of cosine is minus sine. Sine and cosine are similar functions, only shifted by π/2 on the x-axis. They are therefore similar to their own derivatives; this is a relationship with the exponential function ex, which is equal to its derivative. A further connection between the two numbers π and e arises from the antiderivatives of some trigonometric functions, where the natural logarithm ln appears, i.e., the logarithm to the base e.
The integral is the area between the x-axis and a function and, at the same time, the inverse of the derivative. The function of the integral is called the antiderivative. The derivative of an antiderivative is therefore the function itself. Integrating a function is considerably more complicated than differentiating it, and for many functions, an algebraic representation of a term of the antiderivative is not even possible. However, for the simple trigonometric and hyperbolic functions listed here, it is possible.
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.
