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29 results for “derivative”

Formulae (29)

Power Rule (Derivative)

\frac{d}{dx}x^n = n x^{n-1}

Derivative of a power function.

CalculusIntermediate

Chain Rule

\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)

Derivative of a composite function.

CalculusIntermediate

Product Rule

(fg)' = f'g + fg'

Derivative of a product of two functions.

CalculusIntermediate

Quotient Rule

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

Derivative of a quotient of two functions.

CalculusIntermediate

Derivative of Sine

\frac{d}{dx}\sin x = \cos x

Derivative of the sine function.

CalculusIntermediate

Derivative of Cosine

\frac{d}{dx}\cos x = -\sin x

Derivative of the cosine function.

CalculusIntermediate

Derivative of e^x

\frac{d}{dx}e^x = e^x

The exponential function is its own derivative.

CalculusIntermediate

Derivative of ln x

\frac{d}{dx}\ln x = \frac{1}{x}

Derivative of the natural logarithm.

CalculusIntermediate

Power Rule (Integral)

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \; n \ne -1

Antiderivative of a power function.

CalculusAdvanced

Derivative (Limit Definition)

f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}

Formal definition of the derivative.

CalculusBasic

Constant Rule (Derivative)

\frac{d}{dx}(c) = 0

The derivative of any constant is zero.

CalculusBasic

Sum Rule (Derivative)

(f+g)' = f' + g'

The derivative of a sum is the sum of the derivatives.

CalculusBasic

Constant Multiple Rule

\frac{d}{dx}(c\,f) = c\,f'

Constants factor out of a derivative.

CalculusIntermediate

Second Derivative

f''(x) = \frac{d}{dx}\left(f'(x)\right)

The derivative of the first derivative; measures concavity.

CalculusIntermediate

Derivative of Tangent

\frac{d}{dx}\tan x = \sec^2 x

Derivative of the tangent function.

CalculusBasic

Power Rule

\frac{d}{dx}x^n = nx^{n-1}

Derivative of a power function.

CalculusIntermediate

Product Rule

(uv)' = u'v + uv'

Derivative of a product of two functions.

CalculusIntermediate

Quotient Rule

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Derivative of a quotient of two functions.

CalculusIntermediate

Chain Rule

\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}

Derivative of a composite function.

CalculusBasic

Derivative of Sine

\frac{d}{dx}\sin x = \cos x

The derivative of sine is cosine.

CalculusBasic

Derivative of Cosine

\frac{d}{dx}\cos x = -\sin x

The derivative of cosine is negative sine.

CalculusIntermediate

Derivative of Exponential

\frac{d}{dx}e^x = e^x

The exponential function is its own derivative.

CalculusIntermediate

Derivative of Natural Log

\frac{d}{dx}\ln x = \frac{1}{x}

The derivative of the natural logarithm.

CalculusBasic

Power Rule for Integration

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

Antiderivative of a power function (n≠-1).

CalculusIntermediate

Integral of Exponential

\int e^x\,dx = e^x + C

Antiderivative of the exponential function.

CalculusIntermediate

Integral of 1/x

\int \frac{1}{x}\,dx = \ln|x| + C

Antiderivative of the reciprocal function.

CalculusIntermediate

Second Derivative Test

f''(x) > 0 \Rightarrow \text{min}

Concavity determines local minima and maxima.

CalculusAdvanced

Partial Derivative

\frac{\partial f}{\partial x}

Rate of change with respect to one variable, holding others fixed.

CalculusAdvanced

Gradient Vector

\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)

Vector of partial derivatives pointing in the steepest ascent.