Derivative Calculator
Differentiates term by term and names the rule used for each one. Handles polynomials, including negative powers, plus the standard functions.
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.
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 a function needs the chain rule, which this calculator does not apply.
Questions
Asked often enough to answer here
answers that change how you work
Does it do the chain rule?
Not yet. It differentiates term by term, so a term like sin(2x) or (x + 1)³ is out of scope. Terms of the form k·f(x) are fine.
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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