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Examples for

Derivatives

Derivatives measure the rate of change along a curve with respect to a given real or complex variable. Wolfram|Alpha is a great resource for determining the differentiability of a function, as well as calculating the derivatives of trigonometric, logarithmic, exponential, polynomial and many other types of mathematical expressions. Differentiation has many applications within physics, trigonometry, analysis, optimization and other fields.

Derivatives

Differentiate an expression with respect to a given variable.

Calculate the derivative of a function:

derivative of x^4 sin x

differentiate erf(t)^2 wrt t

Start Derivative, Start independent variable, θ , independent variable End,Start function, Start Power, Start base, Start Sine, Start angle, θ , angle End , Sine End , base End,Start exponent, 2 , exponent End , Power End , function End , Derivative Endddθ sinθ2

Implicit Differentiation

Differentiate functions implicitly defined by equations.

Differentiate an equation:

differentiate x^2 - 4y^2 = 1 with respect to x

Compute a derivative using implicit differentiation:

find dy/dx given x^3 - 3 x^2 y +2 x y^2 = 12

Derivatives of Abstract Functions

Find the derivative of an arbitrary function.

Compute derivatives involving abstract functions:

d/dx f(x)+g(x)+h(x)

d/dx[ x f(x^2) ]

Compute partial derivatives of abstract functions:

d/dy f(x^2 + x y +y^2)

Higher-Order Derivatives

Calculate higher-order derivatives.

Compute higher-order derivatives:

second derivative of sin(2x)

d^4/dt^4(Ai(t))

Start Second Order Derivative, Start independent variable, t , independent variable End,Start function, Start Exponential, Start exponent, - Start Power, Start base, t , base End,Start exponent, 2 , exponent End , Power End , exponent End , Exponential End , function End , Second Order Derivative Endd2dt2 ⅇ-t2

Partial Derivatives

Find the partial derivative with respect to a single variable or compute mixed partial derivatives.

Compute partial derivatives:

d/dx x^2 y^4, d/dy x^2 y^4

Compute higher-order partial derivatives:

Start Mixed Partial Derivative, Start first variable, x , first variable End,Start second variable, y , second variable End,Start function, Start Power, Start base, x , base End,Start exponent, 2 , exponent End , Power End Start Power, Start base, y , base End,Start exponent, 4 , exponent End , Power End , function End , Mixed Partial Derivative End∂2∂x∂y x2y4

d^2/dθ^2 (r cos(θ))

Differentiability

Check if functions are differentiable over the field of real numbers.

Test differentiability of a function:

is f(x) = sin^2(x) differentiable?

does abs(x) have a derivative?

is 3xy^2 - x^3 a differentiable function?

{cos(x), sin(x)}: is it differentiable?

GO FURTHER

Step-by-Step Solutions for Calculus

Calculus Web App

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Applications of Calculus

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Directional Derivatives

Calculate the derivative of a multivariate function in a specified direction.

Compute a directional derivative:

derivative of x^2 y+ x y^2 in the direction (1,1)

Derivative Applications

Explore many applications of derivatives.

Find intervals of monotonicity:

where is 300 x - x^3 increasing?

Explore notable points of curves:

stationary points of cos^2(x)

points of inflection of x^3 - 9

Compute extreme values of univariate and multivariate functions:

local extrema of x^2 e^(1/x)

maximize x y on x^2+2 y^2=4

Calculate the tangent line of an equation at a given point:

tangent to y = 3x^4 + 7x^2 - 9 at x = 4