Derivative Calculator

What the rules actually are

The derivative of a function, written f′(x) or dy/dx, is its rate of change with respect to x, which is the slope of the tangent to its graph at each point. Differentiation is not a search for a clever trick. It is a short list of rules applied from the outside of an expression inwards, and the whole skill is recognising which shape you are looking at before you start. There are six that do almost all the work.

• Sum and difference. Differentiate each term on its own. x³ − 3x becomes 3x² − 3 because the two terms never interact.
• Power rule. xⁿ becomes n·x^(n−1). It holds for every constant exponent, negative and fractional included, so 1/x written as x⁻¹ gives −x⁻².
• Constant multiple. A number in front comes along for the ride. 3x gives 3, and you do not need the product rule to see it.
• Product rule. For two things that both vary, u′v + uv′. Two terms, not one, which is the mistake this rule exists to prevent.
• Quotient rule. (u′v − uv′) / v². The order of the numerator matters and the minus sign is where most marks are lost.
• Chain rule. A function of a function multiplies by the derivative of the inside. sin(x²) gives cos(x²)·2x, and the 2x is the part people forget.

Why the calculator names the rule

An answer on its own is only useful if you already trusted it. The reason this tool lists which rule it applied at each step is that the rule is the part you are being examined on, and the part you can check against your own working line by line. If your answer disagrees with the one here, the step list tells you where it diverged rather than only that it did.

The working reads from the outside in, which is the order the rules are genuinely applied. For exp(−x²/2) the outermost shape is a function of a function, so the chain rule goes first; only then does the inside −x²/2 need the constant multiple rule and the power rule. Indentation in the step list shows which part of the expression each rule was looking at.

Worked example

Take x² · cos(3x). Two factors that both vary, so it is the product rule:

• Differentiate x² by the power rule: 2x.
• Differentiate cos(3x) by the chain rule: the outside gives −sin(3x), the inside gives 3, so −3·sin(3x).
• Assemble u′v + uv′: 2x·cos(3x) − x²·3·sin(3x).

Notice the sign. Nothing was subtracted anywhere in the product rule; the minus arrived from the derivative of cosine and then travelled to the front of the second term. That is the kind of detail a bare answer hides.

Where a derivative does not exist

Some functions have no gradient at particular points, and this tool refuses them rather than printing something plausible. abs(x) has a corner at zero: approach from the left and the slope is −1, from the right it is +1, so there is no single answer. The value usually quoted for it is sign(x), which is fine away from zero and wrong at exactly the point anybody studying corners is looking at.

The same applies to floor, ceil, round and trunc, which are flat between jumps and undefined at them, and to min and max, which bend where their arguments cross. Refusing with a reason is more useful than an answer that is correct almost everywhere.

Common mistakes

• Treating 2ˣ with the power rule. The power rule needs a constant exponent. With the variable in the exponent the answer is 2ˣ·ln 2, not x·2^(x−1).
• Forgetting the inside derivative. The chain rule is the most commonly dropped factor on the list. sin(3x) differentiates to 3·cos(3x), not cos(3x).
• Getting the quotient numerator backwards. u′v − uv′, in that order. Swapping the terms flips the sign of the whole answer.
• Reaching for the product rule on a constant multiple. It gives the right answer and signals that you have not spotted the simpler route, which costs time in an exam.
• Assuming log means natural log. Here log is base ten and ln is natural, so their derivatives differ by a factor of ln 10.