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
d/dx(e^(ax))
differentiate erf(t)^2 wrt t
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Higher-Order Derivatives

Calculate higher-order derivatives.

Compute higher-order derivatives:

second derivative of sin(2x)
d^4/dt^4(Ai(t))
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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
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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:

d^2/dθ^2 (r cos(θ))
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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)
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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)
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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?
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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
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