Online Derivative Calculator

What are derivatives?

The derivative is an important tool in calculus that represents an infinitesimal change in a function with respect to one of its variables.

Given a function f (x)f x, there are many ways to denote the derivative of ff with respect to xx. The most common ways are Start Fraction, Start numerator, d f , numerator End,Start denominator, d x , denominator End , Fraction Endd fd x and f'(x)f'x. When a derivative is taken nn times, the notation Start Fraction, Start numerator, Start Power, Start base, d , base End,Start exponent, n , exponent End , Power End f , numerator End,Start denominator, d Start Power, Start base, x , base End,Start exponent, n , exponent End , Power End , denominator End , Fraction Enddn fdxn or Start Power, Start base, f , base End,Start exponent, n , exponent End , Power End (x)fnx is used. These are called higher-order derivatives. Note for second-order derivatives, the notation f''(x)f''x is often used.

At a point x = ax = a, the derivative is defined to be f'(a) = Start Limit, Start variable, h , variable End,Start target value, 0 , target value End,Start expression, Start Fraction, Start numerator, f (a + h) - f (h) , numerator End,Start denominator, h , denominator End , Fraction End , expression End , Limit Endf'a = limhmm-template-arrow-right-80f a + h - f hh . This limit is not guaranteed to exist, but if it does, f (x)f x is said to be differentiable at x = ax = a. Geometrically speaking, f'(a)f'a is the slope of the tangent line of f (x)f x at x = ax = a.

As an example, if f (x) = Start Power, Start base, x , base End,Start exponent, 3 , exponent End , Power Endf x = x3, then f'(x) = Start Limit, Start variable, h , variable End,Start target value, 0 , target value End,Start expression, Start Fraction, Start numerator, Start Power, Start base, (h+x) , base End,Start exponent, 3 , exponent End , Power End - Start Power, Start base, x , base End,Start exponent, 3 , exponent End , Power End , numerator End,Start denominator, h , denominator End , Fraction End , expression End , Limit End = 3 Start Square, Start base, x , base End , Square Endf'x = limhmm-template-arrow-right-80h+x3-x3h = 3x2 and then we can compute f''(x)f''x: f''(x) = Start Limit, Start variable, h , variable End,Start target value, 0 , target value End,Start expression, Start Fraction, Start numerator, 3 Start Power, Start base, (x+h) , base End,Start exponent, 2 , exponent End , Power End -3 Start Power, Start base, x , base End,Start exponent, 2 , exponent End , Power End , numerator End,Start denominator, h , denominator End , Fraction End , expression End , Limit End = 6xf''x = limhmm-template-arrow-right-803x+h2-3 x2h = 6x. The derivative is a powerful tool with many applications. For example, it is used to find local/global extrema, find inflection points, solve optimization problems and describe the motion of objects.