Derivative Calculator
Calculate derivatives of polynomial, trigonometric, exponential, and logarithmic functions with step-by-step solutions. Perfect for calculus students, teachers, and professionals.
What Is a Derivative?
In calculus, the derivative measures how a function changes as its input changes. Geometrically, the derivative of a function at a point is the slope of the tangent line to the graph of the function at that point. The derivative is a fundamental concept in calculus with applications in physics, engineering, economics, and many other fields.
The derivative of a function f(x) is denoted as f'(x) or dy/dx. It is defined as:
f'(x) = limh→0 (f(x+h) − f(x)) / h
Common differentiation rules include:
- Power Rule: d/dx(xn) = n·xn−1
- Product Rule: d/dx(f·g) = f'·g + f·g'
- Chain Rule: d/dx(f(g(x))) = f'(g(x))·g'(x)
- Trigonometric: d/dx(sin x) = cos x, d/dx(cos x) = -sin x
- Exponential: d/dx(ex) = ex, d/dx(ax) = ax·ln(a)
- Logarithmic: d/dx(ln x) = 1/x
How Does the Derivative Calculator Work?
The calculator performs symbolic differentiation using a set of rules:
Step 1: Parse the input function into a structured representation.
Step 2: Apply differentiation rules recursively to each term.
Step 3: Simplify the resulting expression.
Step 4: Display the derivative and, optionally, the step-by-step solution.
The calculator supports:
- Polynomials: xn terms with coefficients
- Trigonometric: sin(x), cos(x), tan(x)
- Exponential: exp(x) or ex
- Logarithmic: ln(x)
- Products: Multiplication of functions
- Composition: Nested functions (chain rule)
Why Use This Derivative Calculator?
- Step-by-Step Solutions: Learn the differentiation process.
- Visualization: See the function and its derivative plotted together.
- Multiple Functions: Supports polynomials, trig, exponential, and logarithmic functions.
- Free & Private: No registration, no data storage.
- Mobile-Friendly: Works on any device.
Common Applications of Derivatives
- Physics: Velocity and acceleration are derivatives of position.
- Economics: Marginal cost and revenue are derivatives of total cost and revenue.
- Optimization: Finding maximum and minimum values of functions.
- Machine Learning: Gradient descent uses derivatives to minimize loss functions.
- Biology: Population growth rates are derivatives of population functions.
❓ Derivative Calculator FAQ
What is a derivative?
A derivative measures how a function changes as its input changes. It represents the slope of the tangent line to the function's graph at any point.
What functions does this calculator support?
This calculator supports polynomials (x^n), trigonometric functions (sin, cos, tan), exponential functions (e^x), and logarithmic functions (ln x). It also handles products and compositions of these functions.
What is the power rule?
The power rule states that d/dx(x^n) = n·x^(n-1). For example, the derivative of x^3 is 3x^2.
What is the product rule?
The product rule states that d/dx(f·g) = f'·g + f·g'. It is used when differentiating the product of two functions.
What is the chain rule?
The chain rule states that d/dx(f(g(x))) = f'(g(x))·g'(x). It is used when differentiating composite functions.
How do I enter a function?
Use standard mathematical notation. For example, enter "x^3 + 2*x^2 - 5*x + 3" for a polynomial. Use "sin(x)", "cos(x)", "tan(x)", "exp(x)", or "ln(x)" for special functions.
What does the derivative represent geometrically?
Geometrically, the derivative at a point represents the slope of the tangent line to the function's graph at that point. It indicates how steeply the function is rising or falling.
What is the difference between f'(x) and dy/dx?
Both notations represent the derivative. f'(x) is the Lagrange notation, while dy/dx is the Leibniz notation. They mean the same thing: the derivative of f with respect to x.
What is the derivative of sin(x)?
The derivative of sin(x) is cos(x). This calculator can compute this and other trigonometric derivatives.
What is the derivative of e^x?
The derivative of e^x is e^x. This is a unique property of the exponential function — it is its own derivative.
What is the derivative of ln(x)?
The derivative of ln(x) is 1/x, for x > 0.
What is the derivative of tan(x)?
The derivative of tan(x) is sec²(x) = 1/cos²(x).
How do I evaluate the derivative at a specific point?
Enable the "Show advanced options" toggle and enter a value in the "Evaluate at x =" field. The calculator will evaluate the derivative at that point.
Why is the chain rule important?
The chain rule allows us to differentiate composite functions — functions within functions. It is one of the most important rules in calculus and is used extensively in optimization and machine learning.
What is a higher-order derivative?
A higher-order derivative is the derivative of a derivative. The second derivative f''(x) measures the rate of change of the first derivative, indicating concavity. This calculator focuses on first derivatives.
How accurate is this derivative calculator?
This calculator performs symbolic differentiation using precise mathematical rules. The results are exact for the supported function types. Numeric evaluations are rounded to the specified precision.
What are derivatives used for in real life?
Derivatives are used in physics (velocity, acceleration), economics (marginal analysis), engineering (optimization), biology (population dynamics), machine learning (gradient descent), and many other fields.
Can I use this calculator for homework?
Yes! This calculator is designed to help students learn calculus by showing step-by-step solutions. It's a great tool for checking your work and understanding the differentiation process.