What is a derivative in calculus?

A derivative measures the instantaneous rate of change of a function. Geometrically, it represents the slope of the tangent line to a curve at a given point. If f(x) describes position over time, f'(x) gives velocity — how fast the position is changing at each moment.

How do I calculate a derivative step by step?

Apply differentiation rules in order: (1) identify the outermost operation (sum, product, quotient, composition), (2) apply the corresponding rule (sum rule, product rule, quotient rule, chain rule), (3) differentiate each inner part recursively, and (4) simplify the result. This calculator shows each of these steps automatically.

What is the power rule?

The power rule states that d/dx[x^n] = n · x^(n-1). For example, the derivative of x^5 is 5x^4. It works for any real exponent n, including fractions (for roots) and negative numbers (for reciprocal powers).

What is the chain rule?

The chain rule differentiates composite functions: d/dx[f(g(x))] = f'(g(x)) · g'(x). You differentiate the outer function, keep the inner function unchanged, then multiply by the derivative of the inner function. For example, d/dx[sin(x²)] = cos(x²) · 2x.

What is the product rule?

The product rule differentiates products of two functions: d/dx[f(x) · g(x)] = f'(x) · g(x) + f(x) · g'(x). For example, d/dx[x² · sin(x)] = 2x · sin(x) + x² · cos(x).

What is the quotient rule?

The quotient rule differentiates fractions: d/dx[f(x)/g(x)] = [f'(x) · g(x) - f(x) · g'(x)] / [g(x)]². A helpful mnemonic is "low d-high minus high d-low, over the square of what's below."

What is a second derivative?

The second derivative f''(x) is the derivative of the derivative. It measures how the rate of change itself is changing — in physics, this is acceleration. A positive second derivative means the function is concave up (bowl shape), while a negative second derivative means concave down.

What functions can this calculator differentiate?

This calculator handles polynomials, trigonometric functions (sin, cos, tan, sec, csc, cot), inverse trig (arcsin, arccos, arctan), exponentials (e^x, a^x), logarithms (ln, log10), hyperbolic functions (sinh, cosh, tanh), square roots, and any combination using sum, product, quotient, power, and chain rules.

How do I find the derivative at a specific point?

Use the "At a Point" mode: enter your function and the x-value where you want to evaluate the derivative. The calculator computes the symbolic derivative first, then substitutes the x-value to give you the numerical slope at that point.

What is the difference between a derivative and an integral?

Differentiation and integration are inverse operations (Fundamental Theorem of Calculus). A derivative finds the rate of change of a function, while an integral finds the accumulated area under a curve. If F'(x) = f(x), then ∫f(x)dx = F(x) + C.

Derivative Calculator

f(x) =

= x^3 + 2x^2 - 5x + 3

Variable

Examples

Polynomial

Trigonometric

Exponential

Logarithmic

Quotient

Chain Rule

Product

Inverse Trig

Square Root

Power of Trig

Hyperbolic

Composite

Derivative

Sum Rule

3x^2 + 2 · 2x - 5

Graph

f(x) and f'(x)

f(x)

f'(x)

Step-by-Step Solution

Differentiation steps applied

Power Rule

d/dx[u^n] = n · u^(n-1) · u'

u = x, n = 3

3x^2

Power Rule

d/dx[u^n] = n · u^(n-1) · u'

u = x, n = 2

Constant Multiple Rule

d/dx[c · f] = c · f'

c = 2, f' = 2x

2 · 2x

Sum Rule

d/dx[f + g] = f' + g'

f' = 3x^2, g' = 2 · 2x

3x^2 + 2 · 2x

Identity Rule

d/dx[x] = 1

Constant Multiple Rule

d/dx[c · f] = c · f'

c = 5, f' = 1

Difference Rule

d/dx[f - g] = f' - g'

f' = 3x^2 + 2 · 2x, g' = 5

3x^2 + 2 · 2x - 5

Constant Rule

d/dx[c] = 0

c = 3

Sum Rule

d/dx[f + g] = f' + g'

f' = 3x^2 + 2 · 2x - 5, g' = 0

3x^2 + 2 · 2x - 5

Common Derivative Rules

Quick reference for standard differentiation formulas

x^nn · x^(n-1)

sin(x)cos(x)

cos(x)-sin(x)

tan(x)sec²(x)

e^xe^x

ln(x)1/x

a^xa^x · ln(a)

√x1/(2√x)

arcsin(x)1/√(1-x²)

arctan(x)1/(1+x²)

What Is a Derivative?

The fundamental concept of calculus that measures rate of change

A derivative measures how a function changes as its input changes. Formally, the derivative of f(x) at a point is defined as the limit of the difference quotient: f'(x) = lim(h→0) [f(x+h) - f(x)] / h. It represents the slope of the tangent line to the function at that point.

Derivatives are used throughout science and engineering to model velocity, acceleration, rates of reaction, marginal cost, and optimization problems. The notation f'(x), dy/dx, and Df(x) all refer to the derivative.

Limit Definition

f'(x) = lim_h→0 [f(x + h) − f(x)] / h

Leibniz

dy/dx

Lagrange

f'(x)

Newton

ẏ

Key Differentiation Rules

The building blocks for computing any derivative

Power Rule

d/dx[xⁿ] = n · xⁿ⁻¹

Chain Rule

d/dx[f(g(x))] = f'(g(x)) · g'(x)

Product Rule

d/dx[f · g] = f'g + fg'

Quotient Rule

d/dx[f/g] = (f'g − fg') / g²

Exponential Rule

d/dx[eˣ] = eˣ

Logarithmic Rule

d/dx[ln(x)] = 1/x

Trigonometric

d/dx[sin(x)] = cos(x)

Inverse Trig

d/dx[arctan(x)] = 1/(1+x²)

This calculator applies these rules automatically, including combinations like the chain rule nested inside a product rule. It supports polynomials, trigonometric, exponential, logarithmic, inverse trigonometric, and hyperbolic functions.

Worked Examples

Step-by-step walkthroughs of common derivative problems

Polynomial: d/dx[x³ + 2x² − 5x + 3]

Apply power rule to each term:

3x² + 2·2x¹ − 5·1 + 0

= 3x² + 4x − 5

Chain Rule: d/dx[sin(x²)]

Outer: d/du[sin(u)] = cos(u)

Inner: d/dx[x²] = 2x

Multiply: cos(x²) · 2x

= 2x · cos(x²)

Product Rule: d/dx[x² · eˣ]

f = x², f' = 2x

g = eˣ, g' = eˣ

f'g + fg' = 2x·eˣ + x²·eˣ

= eˣ(2x + x²)

Quotient Rule: d/dx[x/(x² + 1)]

f = x, f' = 1, g = x²+1, g' = 2x

(f'g − fg') / g² = (1·(x²+1) − x·2x) / (x²+1)²

= (1 − x²) / (x² + 1)²

Higher-Order Derivatives

Understanding second, third, and nth derivatives

The second derivative f''(x) measures the rate of change of the rate of change — in physics, this is acceleration. If f''(x) > 0, the function is concave up; if f''(x) < 0, it is concave down.

Higher-order derivatives provide increasingly detailed information about a function's behavior. They appear in Taylor series expansions, beam deflection equations, and signal processing. This calculator can compute up to the 10th derivative by repeatedly applying differentiation rules.

Taylor Series Expansion

f(x) ≈ f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + f'''(a)(x−a)³/3! + …

1st Derivative

Slope / Velocity

2nd Derivative

Concavity / Acceleration

3rd Derivative

Jerk / Rate of Acceleration

Real-World Applications

How derivatives are used across science, engineering, and economics

Physics

Velocity is the derivative of position; acceleration is the derivative of velocity. Every equation of motion relies on differentiation.

Economics

Marginal cost and marginal revenue are derivatives of total cost and total revenue functions, used to find profit-maximizing output.

Engineering

Signal processing uses derivatives to detect edges, analyze frequency response, and design control systems (PID controllers).

Biology

Population growth rates, enzyme kinetics (Michaelis-Menten), and neural firing rates are all modeled with derivatives.

Machine Learning

Gradient descent — the backbone of neural network training — computes partial derivatives (gradients) to minimize loss functions.

Finance

The Greeks (delta, gamma, theta) in options pricing are derivatives of the Black-Scholes formula with respect to price, time, and volatility.

Common Differentiation Mistakes

Errors to watch for when computing derivatives

Forgetting the Chain Rule

d/dx[sin(3x)] = 3cos(3x), not just cos(3x). Always multiply by the inner derivative.

Wrong Power Rule on eˣ

d/dx[eˣ] = eˣ, not x · eˣ⁻¹. The power rule only applies to xⁿ, not aˣ.

Quotient Rule Sign Error

It's (f'g − fg')/g², not (fg' − f'g). The order matters — "low d-high minus high d-low."

Confusing d/dx with ∂/∂x

For single-variable functions, d/dx treats everything else as constant. Partial derivatives (∂) are for multivariable functions.

Frequently Asked Questions

Common questions about derivatives and differentiation

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Last updated Apr 1, 2026

Derivative Calculator

Graph

Step-by-Step Solution

Common Derivative Rules

What Is a Derivative?

Key Differentiation Rules

Worked Examples

Higher-Order Derivatives

Real-World Applications

Common Differentiation Mistakes

Frequently Asked Questions

Embed Derivative Calculator

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