How does the chain rule work?

The derivative of a composite function f(g(x)) is the outer derivative evaluated at the inner function times the derivative of the inner function: [f(g(x))]′ = f′(g(x))·g′(x). For sin(x²) that gives cos(x²)·2x. In the steps you’ll see “Chain rule” with the outer derivative cos(u), followed by a separate step for the inner function u = x².

How do you differentiate xˣ?

When both base and exponent contain x, neither the power rule nor the exponential rule applies directly. Use logarithmic differentiation: let y = xˣ, take logs to get ln y = x·ln x, differentiate to get y′/y = ln x + 1, so y′ = xˣ(ln x + 1). The calculator handles every function of this kind the same way.

Why doesn’t my answer look like the textbook’s?

One derivative can be written in many equivalent forms. The calculator combines like terms and powers of the same base but does not put everything over a common denominator, factor or apply trig identities, so (x·cos x − sin x)/x² and cos x/x − sin x/x² are both possible. Use “Evaluate at a point” to compare the two forms numerically — if the values agree, both are correct.

How do I get partial and higher-order derivatives?

When the expression has several letter variables (e and pi are constants), a “with respect to” selector appears; the other variables are treated as constants and the full gradient is listed below. Choose an order from 2 to 10 for repeated differentiation, and expand the steps for each order. Note that xy is read as x·y.

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∂Derivative Calculator With Steps - Higher Order & Partial

This calculator differentiates functions symbolically and shows which rule is used at every step: the sum and constant multiple rules, the product rule (uv)′ = u′v + uv′, the quotient rule (u/v)′ = (u′v − uv′)/v², the power rule, the chain rule, the standard derivatives of exponentials, ln and logₐ, trigonometric, inverse trigonometric and hyperbolic functions, and logarithmic differentiation for functions like xˣ or x^sin x where both base and exponent depend on x. Each intermediate result is simplified by folding constants, removing 0 and 1, combining like terms and merging powers of the same base. You can take derivatives up to order 10; with several variables, as in x²y + sin(xy), pick the variable for a partial derivative and see the whole gradient. At any point you choose, the function and each derivative are evaluated and checked against a central finite difference, so you can trust the symbolic answer. For a single variable, one click plots f and its derivatives in the graphing calculator.

Free · No sign-up · Runs in your browserLast updated: 2026-10-26

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How to use Derivative Calculator

1Enter a function such as x^2*sin(x), ln(x^2+1) or x^x, or pick an example; use x, y, z for several variables
2Choose the order, and for several variables the one to differentiate with respect to
3Read the result and the step-by-step working, evaluate at a point, or open the graph of f and f′

FAQ

How does the chain rule work?: The derivative of a composite function f(g(x)) is the outer derivative evaluated at the inner function times the derivative of the inner function: [f(g(x))]′ = f′(g(x))·g′(x). For sin(x²) that gives cos(x²)·2x. In the steps you’ll see “Chain rule” with the outer derivative cos(u), followed by a separate step for the inner function u = x².
How do you differentiate xˣ?: When both base and exponent contain x, neither the power rule nor the exponential rule applies directly. Use logarithmic differentiation: let y = xˣ, take logs to get ln y = x·ln x, differentiate to get y′/y = ln x + 1, so y′ = xˣ(ln x + 1). The calculator handles every function of this kind the same way.
Why doesn’t my answer look like the textbook’s?: One derivative can be written in many equivalent forms. The calculator combines like terms and powers of the same base but does not put everything over a common denominator, factor or apply trig identities, so (x·cos x − sin x)/x² and cos x/x − sin x/x² are both possible. Use “Evaluate at a point” to compare the two forms numerically — if the values agree, both are correct.
How do I get partial and higher-order derivatives?: When the expression has several letter variables (e and pi are constants), a “with respect to” selector appears; the other variables are treated as constants and the full gradient is listed below. Choose an order from 2 to 10 for repeated differentiation, and expand the steps for each order. Note that xy is read as x·y.