Derivative Calculator

What is a Derivative?

A derivative measures the instantaneous rate of change of a function with respect to its variable. Geometrically, the derivative at a point represents the slope of the tangent line to the function's graph at that point. Derivatives are fundamental to calculus and have widespread applications in physics, engineering, economics, and many other fields.

The derivative of a function $f(x)$ with respect to $x$ is defined as the limit:

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

How to Use the Derivative Calculator

Our comprehensive derivative calculator supports four types of derivatives to handle all your calculus needs:

• Single Variable Derivative: Compute derivatives of functions with one variable, with orders from 1 to 10. The calculator identifies which differentiation rules are applied and shows each step.
• Partial Derivative: For functions of multiple variables, partial derivatives measure the rate of change with respect to one variable while treating others as constants. Supports mixed partial derivatives.
• Implicit Derivative: Finds derivatives when a function is defined implicitly by an equation $F(x, y) = 0$ without explicitly solving for $y$.
• Directional Derivative: Measures the rate of change of a function in any specified direction by computing the gradient and taking its dot product with the unit direction vector.

Common Differentiation Rules

The derivative calculator applies these fundamental rules when computing derivatives:

• Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$. Example: $\frac{d}{dx}[x^3] = 3x^2$.
• Product Rule: $\frac{d}{dx}[f \cdot g] = f'g + fg'$. Example: $\frac{d}{dx}[x \cdot \sin(x)] = \sin(x) + x\cos(x)$.
• Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$. Example: $\frac{d}{dx}[\sin(x^2)] = 2x\cos(x^2)$.
• Quotient Rule: $\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$.
• Sum Rule: $\frac{d}{dx}[f + g] = f' + g'$.
• Constant Rule: $\frac{d}{dx}[c] = 0$.

Function Input Syntax

When entering functions, use the following syntax:

• Exponents: Use ^ (e.g., x^2 for x squared, x^3 for x cubed)
• Multiplication: Use * (e.g., 2*x, x*y) - implicit multiplication like 2x also works
• Trigonometric: sin(x), cos(x), tan(x), cot(x), sec(x), csc(x)
• Inverse trig: asin(x), acos(x), atan(x)
• Exponential: exp(x) or e^x
• Logarithms: ln(x) for natural log, log(x, base) for other bases
• Square root: sqrt(x) or x^(1/2)
• Absolute value: abs(x)

Partial Derivative Formula

For a multivariable function $f(x, y, z, ...)$, the partial derivative with respect to $x$ treats all other variables as constants. The notation $\frac{\partial f}{\partial x}$ indicates the partial derivative with respect to $x$. Mixed partial derivatives like $\frac{\partial^2 f}{\partial x \partial y}$ compute the rate of change sequentially.

Implicit Differentiation

For an equation $F(x, y) = 0$ defining $y$ implicitly as a function of $x$, the implicit derivative is computed using the formula:

$$\frac{dy}{dx} = -\frac{F_x}{F_y}$$

where $F_x = \frac{\partial F}{\partial x}$ and $F_y = \frac{\partial F}{\partial y}$. This formula avoids solving for $y$ explicitly.

Directional Derivative

The directional derivative measures the rate of change of $f$ in the direction of a unit vector $\hat{u}$:

$$D_{\hat{u}} f = \nabla f \cdot \hat{u}$$

where $\nabla f = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, ... \rangle$ is the gradient vector. The direction vector is normalized before computing the dot product.

Applications of Derivatives

• Physics: Velocity is the derivative of position; acceleration is the derivative of velocity. Used in kinematics, electromagnetism, and thermodynamics.
• Engineering: Optimization problems, control systems, signal processing, and structural analysis all rely on derivatives.
• Economics: Marginal cost, revenue, and profit are derivatives. Elasticity of demand uses derivatives for sensitivity analysis.
• Machine Learning: Gradient descent optimization uses derivatives to minimize loss functions during model training.

Also check: Integral Calculator, Limit Calculator, Implicit Differentiation Calculator.