Derivative Calculator
Differentiate a function symbolically, with the rules used and a graph of f and f′
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Finds the derivative of a function symbolically and shows, step by step, which differentiation rules were used. Use it to check homework answers or to find a second derivative, the slope at a point or a tangent line.
How to use
- Type the function into "Function f" the way you write it, for example x^2 sin x, and check that "Read as" shows what you meant.
- If there are several letters, choose the variable under "Differentiate with respect to", and pick 1 to 5 under "Order of the derivative".
- Read the "Derivative" and "Steps: rules applied". Use "Copy as text" or "Copy LaTeX" to copy the result.
- Enter a point under "Value at a point and tangent line" to get the function value, the slope, the tangent line and a graph.
How it works
- The steps name each rule (sum, constant multiple, product, quotient, chain rule, basic derivatives) and give the derivative of each part.
- Every result is checked against numerical differentiation at random points before it is shown; a result that does not agree is not shown.
- Supported: powers and roots, e, pi, ln, log (base 10), trigonometric, inverse trigonometric and hyperbolic functions, and absolute value. Angles are in radians. Factorials and piecewise functions are not supported.
- In an expression with several letters, the other letters are treated as constants, and the first partial derivatives are shown as well.
Questions and answers
- What is the derivative of x^2 sin x?
- By the product rule (fg)′ = f′g + fg′, it is 2x sin x + x² cos x. The example button shows the steps.
- How do I use the chain rule?
- The derivative of f(g(x)) is f′(g(x)) × g′(x). For example, the derivative of sin(x²) is 2x cos(x²).
- How do I find the second derivative?
- Choose 2 under "Order of the derivative". Orders up to 5 are available; if a result gets very long, only the lower orders are shown.
- How do I find the equation of a tangent line?
- The tangent line at x = a is y = f′(a)(x − a) + f(a). Enter a as the point, and the slope and the tangent line are calculated and drawn on the graph.