📉 Derivative Calculator
Differentiate functions step by step, with every rule shown.
f'(x)
How to Use the Derivative Calculator
Pick the function type, enter the coefficients, and calculate. You will get the derivative plus a step-by-step breakdown naming each rule used.
Choose the function family: polynomial, trigonometric, exponential, or logarithmic.
Enter the coefficients. For a·x^n, a is the multiplier and n is the power.
Read the steps: each term shows the rule applied, so you learn while you solve.
Differentiation Rules Reference
Power rule: d/dx[x^n] = n·x^(n−1). The workhorse: x³ becomes 3x². Constant multiple: constants ride along: d/dx[5x²] = 5·2x = 10x. Sum/difference: differentiate each term separately. Trig: d/dx[sin x] = cos x; d/dx[cos x] = −sin x; d/dx[tan x] = sec²x. Exponential: d/dx[e^x] = e^x; d/dx[a^x] = a^x·ln a. Log: d/dx[ln x] = 1/x. Chain rule: for f(g(x)), multiply the outer derivative by g′(x).
Worked Examples
Polynomial: f(x) = 4x³. Power rule: bring down the 3, reduce the exponent: f′(x) = 4·3·x² = 12x². Sum: f(x) = 3x² + 5x − 7. Term by term: 6x + 5 + 0 = 6x + 5. Constants vanish. Chain rule: f(x) = sin(2x). Outer derivative cos(2x), times inner derivative 2: f′(x) = 2cos(2x). Exponential: f(x) = 3e^(2x). Derivative of e^(2x) is 2e^(2x) by the chain rule, times 3: f′(x) = 6e^(2x).
Common Mistakes to Avoid
Forgetting the chain rule's inner derivative is mistake number one: d/dx[(3x+1)²] is 2(3x+1)·3 = 6(3x+1), not 2(3x+1). Mistake two: treating the derivative of a product as the product of derivatives; use the product rule instead. Mistake three: sign errors on cos and tan derivatives. Mistake four: applying the power rule to e^x (it does not apply; e^x is its own derivative). Work slowly and name each rule as you use it.
Frequently Asked Questions
What is a derivative?
A derivative measures how fast a function changes at each point: the slope of its tangent line. If f(x) = x², then f'(x) = 2x, meaning the slope at x=3 is 6. Derivatives power physics (velocity from position), economics (marginal cost), and machine learning (gradient descent).
What are the basic differentiation rules?
Power rule: d/dx[x^n] = n·x^(n-1). Constant multiple: d/dx[c·f] = c·f'. Sum rule: differentiate term by term. Product rule: (fg)' = f'g + fg'. Quotient rule: (f/g)' = (f'g − fg')/g². Chain rule: d/dx[f(g)] = f'(g)·g'. This calculator shows which rule applies to each term.
How do I differentiate e^x and ln(x)?
e^x is special: its derivative is itself, e^x. For a^x, the derivative is a^x·ln(a). The derivative of ln(x) is 1/x, and ln(g(x)) differentiates to g'(x)/g(x) by the chain rule. Select the function type above and enter the pieces.
What is the chain rule?
The chain rule handles functions inside functions: if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). Differentiate the outside, leave the inside alone, then multiply by the derivative of the inside. Example: d/dx[sin(x²)] = cos(x²) × 2x. Forgetting the × 2x is the most common calculus mistake.
What is the difference between first and second derivatives?
The first derivative f' gives slope and rate of change. The second derivative f'' gives concavity: how the slope itself is changing. In physics, if position is f, velocity is f' and acceleration is f''. This calculator returns the first derivative; apply it twice for the second.