Differentiation of Fractions

Derivative Calculator

Function of x

First derivative of the function after x

Input area for the fraction:

f(x) =

\frac{d}{d x} f (x)

\frac{d^{n}}{d x^{n}} f (x)

Plot

Pos1

End

(

)

e^{a x}

sin

cos

tan

e^x

x^a

\frac{a}{x}

asin

acos

atan

x²

√x

a^x

\frac{a}{x + b}

|x|

sinh

cosh

\frac{a x + c}{b x + d}

\frac{a + x}{b + x}

\frac{1 + x}{1 - x}

\frac{1}{a + b x}

\frac{1 + \sqrt{x}}{1 - \sqrt{x}}

\frac{x^{2} - a^{2}}{x^{2} + a^{2}}

\frac{\sin (x)}{\cos (x)}

\frac{x}{e^{x}}

Function	Description
sin(x)	Sine of x
cos(x)	Cosine of x
tan(x)	Tangent of x
asin(x)	arcsine
acos(x)	arccosine of x
atan(x)	arctangent of x
atan2(y, x)	Returns the arctangent of the quotient of its arguments.
cosh(x)	Hyperbolic cosine of x
sinh(x)	Hyperbolic sine of x
pow(a, b)	Power a^b
sqrt(x)	Square root of x
exp(x)	e-function
log(x), ln(x)	Natural logarithm
log(x, b)	Logarithm to base b
log2(x), lb(x)	Logarithm to base 2
log10(x), ld(x)	Logarithm to base 10

more ...

Notations

Notation for Derivatives

$\frac{d}{d x} f (x) = \frac{d f}{d x} (x) = \frac{d f (x)}{d x} = f^{'} (x)$

Quotient Rule

The quotient rule specifies how to treat the quotient of two functions when differentiate.

Derivative rule for fractions:

$\frac{d}{d x} f (x) = \frac{d}{d x} \frac{u (x)}{v (x)} = \frac{v (x) \frac{d}{d x} u (x) - u (x) \frac{d}{d x} v (x)}{v^{2} (x)} = \frac{u^{'} (x) \cdot v (x) - u (x) \cdot v^{'} (x)}{v^{^{2}}}$

Example of the application of the quotient rule

Example for the derivative of a fraction:

${(\frac{x + a}{x + b})}^{'}$

Application of the quotient rule with u=x+a and v=x+b

$= \frac{{(x + a)}^{'} (x + b) - (x + a) {(x + b)}^{'}}{{(x + b)}^{2}}$

Derivative of the terms results u′=1 and v′=1

$= \frac{x + b - (x + a)}{{(x + b)}^{2}}$

After simplification

$= \frac{b - a}{{(x + b)}^{2}}$

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