Derivative

Question 1

What is a derivative? (Definition)

Answer

Mathematicians have defined derivatives by the formula $$ \frac{d}{dx}f = f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} $$

The derivative of a function $ f $ is denoted $ f' $ (with an apostrophe named prime) or $ \frac{d}{dx}f $ where $ d $ is the derivative operator and $ x $ the variable on which to derivate.

The derivative calculation is the inverse operation of primitive calculation (indefinite integral).

Question 2

How to calculate a derivative?

Answer

The derivation calculation (first order derivative) is based mainly on a list of usual derivatives, already calculated and known (see below).

On dCode, the derivative calculator knows all the derivatives, indicate the function and the variables on which to derivate to obtain the result of the derivative computation.

Example: $$ f(x) = x^2+\sin(x) \Rightarrow f'(x) = 2 x+\cos(x) $$

Derivative calculus is often used in physics to compute a velocity.

Question 3

What is the list of common derivatives?

Answer

The more useful derivatives are:

Name	Function	Derivative
constant/number	$$ k \in \mathbb{R} $$	$$ 0 $$
variable	$$ x $$	$$ 1 $$
power n (exponent)	$$ x^n $$	$$ n x^{n-1} $$
negative power	$$ x^{-n} $$	$$ -n x^{-n-1} $$
inverse	$$ \frac{1}{x} $$	$$ -\frac{1}{x^2} $$
inverse power	$$ \frac{1}{x^n} $$	$$ -\frac{n}{x^{n+1}} $$
root	$$ \sqrt{x} $$	$$ \frac {1}{2\sqrt{x}} $$
nth root	$$ \sqrt[n]x $$	$$ \frac{1}{n\sqrt[n]{x^{n-1}}} $$
fractional power	$$ x^{1/n} $$	$$ (1/n)x^{(1/n)-1} $$
natural logarithm	$$ \ln \|x\| $$	$$ \frac{1}{x} $$
logarithm base a	$$ \log_a \|x\| $$	$$ \frac{1}{x \ln a} $$
exponential	$$ e^x $$	$$ e^x $$
exponent x	$$ a^x $$	$$ a^x \ln a $$
sine	$$ \sin(x) $$	$$ \cos(x) $$
cosine	$$ \cos(x) $$	$$ - \sin(x) $$
tangent	$$ \tan(x) $$	$$ \frac{1}{\cos^2(x)} \\ = \sec^2(x) \\ = 1+\tan^2(x) \\ = \frac{2}{1+\cos(2x)} $$
secant	$$ \sec(x) = \frac{1}{\cos(x)} $$	$$ \frac{\tan(x)}{\cos(x)} \\ = \sec(x)\tan(x) \\ = \frac{2\sin(x)}{1+\cos(2x)} $$
cosecant	$$ \csc(x) = \frac{1}{\sin(x)} $$	$$ -\frac{\cos(x)}{\sin^2(x)} \\ = -\cot(x)\csc(x) \\ = \frac{2\cos(x)}{-1+\cos(2x)} $$
cotangent	$$ \cot(x) = \frac{1}{\tan(x)} $$	$$ - \frac{1}{\sin^2(x)} \\ = -1-\cot^2(x) \\ = -\csc^2(x) \\ = \frac{2}{-1+\cos(2x)} $$
arcsine	$$ \arcsin(x) $$	$$ \frac{1}{\sqrt{1-x^2}} $$
arccosine	$$ \arccos(x) $$	$$ -\frac{1}{\sqrt{1-x^2}} $$
arctangent	$$ \arctan(x) $$	$$ \frac{1}{1+x^2} $$
sine hyperbolic	$$ \sinh(x) $$	$$ \cosh(x) $$
cosine hyperbolic	$$ \cosh(x) $$	$$ \sinh(x) $$
tangent hyperbolic	$$ \tanh(x) $$	$$ \frac{1}{\cosh^2(x)} \\ = 1 - \tanh^2(x) $$
cotangent hyperbolic	$$ \coth(x) $$	$$ \frac{-1}{\sinh^2(x)} \\ = 1 - \coth^2(x) $$
arcsinus hyperbolic	$$ \operatorname{arcsinh} x $$	$$ \frac{1}{\sqrt{1+x^2}} $$
arccosinus hyperbolic	$$ \operatorname{arccosh} x $$	$$ \frac{1}{\sqrt{x^2-1}} $$
arctangent hyperbolic	$$ \operatorname{arctanh} x $$	$$ \frac{1}{1-x^2} $$

Question 4

What is the list of compound function derivatives?

Answer

Common derivatives of compound functions to know are:

Name	Compound Function	Derivative
compound function	$$ g \circ f $$	$$ (g' \circ f)\times f' $$
function power n (exponentiation)	$$ f^n $$	$$ n f^{n - 1} f' $$
sine of function	$$ \sin(f) $$	$$ f' \cos(f) $$
cosine of function	$$ \cos(f) $$	$$ - f' \sin(f) $$
exponential of function	$$ \exp(f) $$	$$ f' \exp(f) $$
square root of function (positive function)	$$ \sqrt{f} $$	$$ \frac{f'}{2\sqrt{f}} $$
logarithm of function (positive function)	$$ \ln(f) $$	$$ \frac{f'}{f} $$

Question 5

How to calculate a second derivative?

Answer

A second derivative is equivalent to compute the derivative twice, for dCode, indicate twice the same variable to get the second derivative.

Second derivative calculation is often used in physics to compute acceleration (derivative of velocity).

Question 6

How to calculate a partial derivative?

Answer

A partial derivative is a derivative that only applies to one variable, leaving the others intact.

On dCode, indicate a single variable if the function has several to obtain a partial derivative.

Question 7

How to calculate a primitive?

Answer

Use the primitive calculator tool available on dCode.

Derivative

Derivative Calculator

Answers to Questions (FAQ)