Derivative Calculator
Differentiates any expression you can type, applying the chain, product and quotient rules wherever they are needed, and naming every rule it used. It parses the expression into a tree first, which is what lets it see inside a composition.
How it works
How the Derivative Calculator works
the part a result on its own leaves out
What a derivative measures
The instantaneous rate of change — how fast the output moves as the input moves. On a graph it is the slope of the tangent at a point, which is why a zero derivative marks a peak or a trough.
How the calculator sees an expression
It parses what you type into a tree before differentiating anything. sin(2x) becomes a sine applied to a product, and that structure is what makes the chain rule possible — a rule that has to know what is inside what cannot be applied to a flat list of terms.
Why the power rule looks the way it does
It falls out of the limit definition. For xⁿ the difference quotient expands and every term but nxⁿ⁻¹ vanishes as the interval shrinks. The rule is a shortcut for a limit you would otherwise compute each time.
Common mistakes
Common mistakes with derivatives
the errors that actually show up, not every possible slip
Dropping the constant term instead of zeroing it
The derivative of 3x² + 7 is 6x. The 7 does not disappear by accident — a constant has zero rate of change, which is the whole reason it goes.
Forgetting the sign on cos
The derivative of cos x is −sin x. The minus sign is the part people lose, and it flips the answer entirely.
Using the power rule on a composite
sin(2x) is not handled by the power rule. Anything with a function inside another needs the chain rule — multiply by the derivative of the inside. Forgetting that factor is the most common error in early calculus.
Questions
Asked often enough to answer here
answers that change how you work
Does it do the chain rule?
Yes, along with the product and quotient rules, nested to any depth. sin(2x), (x + 1)³, x²·sin x and cos x / x are all handled, and the rules used are listed with the answer.
What notation does it use?
f′(x), read “f prime of x”. dy/dx means the same thing and is more common when the variable being changed matters.
Why is the derivative of eˣ itself?
Because e is defined as the base where the rate of growth equals the current value. That property is what singles e out from every other base.
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