Derivative Calculator

Calculate derivatives step-by-step with detailed solutions. Supports polynomials, trigonometric, exponential, and logarithmic functions.

Syntax: Use ^ for powers, * for multiplication, sin(), cos(), tan(), ln(), exp(), sqrt()
Chain Rule
d/dx[f(g(x))] = f'(g(x)) · g'(x)
Product Rule
d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Quotient Rule
d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]²
Implicit Differentiation
For equations like x² + y² = 25

Common Derivative Examples

Polynomial Function
f(x) = x³ + 2x² - 5x + 1
f'(x) = 3x² + 4x - 5
Chain Rule Example
f(x) = sin(x²)
f'(x) = 2x·cos(x²)
Product Rule Example
f(x) = x·sin(x)
f'(x) = sin(x) + x·cos(x)
Quotient Rule Example
f(x) = (x² + 1)/(x - 1)
f'(x) = (x² - 2x - 1)/(x - 1)²
Exponential & Trigonometric
f(x) = eˣ·cos(x)
f'(x) = eˣ(cos(x) - sin(x))
Logarithmic Function
f(x) = ln(x² + 1)
f'(x) = 2x/(x² + 1)

Derivative Rules & Concepts

Derivatives measure the rate of change of a function. They are fundamental in calculus and have applications in physics, engineering, economics, and many other fields.

Basic Derivative Rules

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

Common Function Derivatives

Function Derivative
$\sin(x)$ $\cos(x)$
$\cos(x)$ $-\sin(x)$
$\tan(x)$ $\sec^2(x)$
$e^x$ $e^x$
$\ln(x)$ $\frac{1}{x}$
$\sqrt{x}$ $\frac{1}{2\sqrt{x}}$

Applications of Derivatives

  • Rate of Change: Velocity, acceleration, growth rates
  • Optimization: Finding maximum and minimum values
  • Curve Sketching: Analyzing function behavior
  • Related Rates: How quantities change with respect to each other
  • Linear Approximation: Estimating function values
  • Physics: Motion, forces, electromagnetic fields
  • Economics: Marginal cost, revenue, profit analysis

Calculus Tip:

When finding derivatives, always identify which rule applies first. For complex functions, break them down into simpler parts and apply the chain rule systematically.

Derivative Calculator Answers

What does the derivative calculator do?

The derivative calculator computes the derivative of a mathematical function with respect to a variable. It helps analyze how a function changes, which is crucial in calculus and real-world applications.

Can it solve higher-order derivatives?

Yes, the calculator can compute first, second, or even higher-order derivatives. Simply specify the order, and the tool will return the result accordingly.

Does it support functions with multiple variables?

Absolutely. The calculator can handle multivariable functions and compute partial derivatives with respect to a chosen variable, making it useful for advanced calculus problems.

Is the calculator suitable for learning purposes?

Yes. It not only provides the final derivative but often shows the steps involved, helping students understand the differentiation process and learn calculus more effectively.

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