Ultimate Derivatives Calculator

Master the art of differentiation with our comprehensive calculator suite. Calculate derivatives of any function, higher-order derivatives, partial derivatives, implicit differentiation, and more with step-by-step solutions.

Derivatives Calculator

Our advanced derivatives calculator handles a wide range of functions including polynomials, trigonometric, logarithmic, exponential, and more. Enter your function below to find its derivative with detailed step-by-step solutions.

Standard Derivatives
Higher-Order Derivatives
Partial Derivatives
Implicit Differentiation
Directional Derivatives

Standard Derivatives Calculator

Calculate the derivative of a function with respect to a variable.

Use x as your variable. For example: x^2, sin(x), e^x, ln(x), etc.
The variable with respect to which the derivative is calculated.

Derivative Result

Step-by-Step Solution

Examples

x^2 + 3x - 5
sin(x) + cos(x)
e^x * ln(x)
(x^2 + 1) / (x - 1)
sqrt(x^2 + 1)

Higher-Order Derivatives Calculator

Calculate second, third, and higher-order derivatives of functions.

Use x as your variable. For example: x^3, sin(x), e^x, etc.
Enter a number between 1 and 10. For example, 2 for second derivative.

Higher-Order Derivative Result

Step-by-Step Solution

Examples

x^3 - 6x^2 + 12x - 8
sin(x)
e^x
ln(x)
x * e^x

Partial Derivatives Calculator

Calculate partial derivatives of multivariable functions with respect to specific variables.

Use variables like x, y, z. For example: x^2 + y^2, sin(x)*cos(y), e^(x+y), etc.
The variable with respect to which the partial derivative is calculated.

Partial Derivative Result

Step-by-Step Solution

Higher-Order Partial Derivatives

Use variables like x, y, z. For example: x^2*y + x*y^2, sin(x*y), etc.
Enter variables in the order of differentiation, separated by commas. For example: x,y for ∂²f/∂y∂x

Mixed Partial Derivative Result

Step-by-Step Solution

Examples

x^2 + 3xy + y^2
sin(x) * cos(y)
e^(x+y)
ln(x^2 + y^2)
x*y*z

Implicit Differentiation Calculator

Calculate derivatives of functions defined implicitly by equations.

Enter an equation in terms of x and y. For example: x^2 + y^2 = 25, x^3 + y^3 = 3*x*y, etc.

Implicit Derivative Result

Step-by-Step Solution

Find Derivative at a Point

Enter an equation in terms of x and y.

Derivative at Point Result

Examples

x^2 + y^2 = 25
x^3 + y^3 = 3xy
sin(x) + cos(y) = xy
e^(xy) = x + y
ln(x) + ln(y) = 1

Directional Derivatives Calculator

Calculate the rate of change of a function in a specific direction.

Enter a function in terms of x, y, and optionally z.
Enter the components of the direction vector. The vector will be normalized automatically.

Directional Derivative Result

Step-by-Step Solution

Examples

x^2 + y^2
xy + yz
e^(x+y)
sin(x) * cos(y)
x^2 + y^2 + z^2

Common Derivative Rules

Understanding the basic rules of differentiation is essential for calculating derivatives. Here are the most important derivative rules with examples:

Power Rule

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

The power rule states that the derivative of x raised to a power equals the power times x raised to the power minus 1.

Example: \frac{d}{dx}(x^3) = 3x^2

Product Rule

\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)

The product rule is used to find the derivative of a product of two functions.

Example: \frac{d}{dx}(x^2 \cdot \sin(x)) = 2x \cdot \sin(x) + x^2 \cdot \cos(x)

Quotient Rule

\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}

The quotient rule is used to find the derivative of a quotient of two functions.

Example: \frac{d}{dx}\left(\frac{x^2}{\sin(x)}\right) = \frac{2x \cdot \sin(x) - x^2 \cdot \cos(x)}{[\sin(x)]^2}

Chain Rule

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

The chain rule is used to find the derivative of a composite function.

Example: \frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot 2x

Exponential Rule

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

The exponential function e^x is its own derivative.

Example: \frac{d}{dx}(e^{3x}) = 3e^{3x}

Logarithmic Rule

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

The derivative of the natural logarithm function is the reciprocal of the input.

Example: \frac{d}{dx}(\ln(2x)) = \frac{1}{2x} \cdot 2 = \frac{1}{x}

Trigonometric Rules

\frac{d}{dx}(\sin(x)) = \cos(x) \\ \frac{d}{dx}(\cos(x)) = -\sin(x) \\ \frac{d}{dx}(\tan(x)) = \sec^2(x)

The derivatives of the basic trigonometric functions.

Example: \frac{d}{dx}(\sin(2x)) = \cos(2x) \cdot 2 = 2\cos(2x)

Inverse Trigonometric Rules

\frac{d}{dx}(\arcsin(x)) = \frac{1}{\sqrt{1-x^2}} \\ \frac{d}{dx}(\arccos(x)) = -\frac{1}{\sqrt{1-x^2}} \\ \frac{d}{dx}(\arctan(x)) = \frac{1}{1+x^2}

The derivatives of the inverse trigonometric functions.

Example: \frac{d}{dx}(\arctan(x^2)) = \frac{1}{1+(x^2)^2} \cdot 2x = \frac{2x}{1+x^4}

Understanding Derivatives and Their Applications

What is a Derivative?

A derivative measures the rate of change of a function with respect to one of its variables. Geometrically, it represents the slope of the tangent line to the function's graph at a given point. The derivative of a function f(x) is denoted as f'(x), df/dx, or d/dx(f(x)).

Types of Derivatives

Applications of Derivatives

Derivatives have numerous applications across various fields:

Mathematics and Calculus

Physics

Engineering

Economics

Computer Science

Common Derivative Formulas

Function f(x) Derivative f'(x)
c (constant) 0
x^n n·x^(n-1)
e^x e^x
a^x a^x·ln(a)
ln(x) 1/x
log_a(x) 1/(x·ln(a))
sin(x) cos(x)
cos(x) -sin(x)
tan(x) sec^2(x)
arcsin(x) 1/√(1-x^2)
arccos(x) -1/√(1-x^2)
arctan(x) 1/(1+x^2)

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