Math calculator
Derivative Calculator
Differentiate any function symbolically, see every rule applied step by step, and graph f(x) against f′(x).
Functions: sin, cos, tan, asin, acos, atan, sqrt (or √), abs (or |x|), ln, log, exp — trig uses radians. The buttons insert into whichever box you clicked last.
Derivative f′(x)
2x
Steps
How it works
Enter f(x) and the calculator differentiates it symbolically — no numerical approximation. It applies the standard differentiation rules to the expression tree and simplifies the result, so d/dx of sin(2x) comes back as 2cos(2x), the way you would write it by hand.
Unlike integration, differentiation always succeeds: the power, product, quotient, and chain rules together cover every function this calculator can parse, including cases like x^x that need logarithmic differentiation. Optionally enter a point to evaluate f′ there — the value is shown exactly (fractions, multiples of π or e) when possible.
Worked example
For f(x) = 3x² − 4x + 5, each term is differentiated with the power rule: d/dx 3x² = 6x, d/dx (−4x) = −4, and the constant 5 vanishes. So f′(x) = 6x − 4, and at x = 2 the slope is 6·2 − 4 = 8.
The graph draws f(x) as a solid curve and f′(x) dashed, so you can see the relationship directly: wherever f has a peak or valley, f′ crosses zero.
Step-by-step solutions
The steps panel under the answer shows the differentiation term by term, naming the rule used for each — power rule, product rule, quotient rule, chain rule, exponential and logarithm rules, inverse trig, and logarithmic differentiation for forms like x^x. When you evaluate at a point, the substitution is written out with the value filled in before the final answer.
Supported syntax
- Operators: + − * / ^ and parentheses, with implicit multiplication (2x, 3sin(x), (x+1)(x−2)).
- Constants: pi (or π) and e — usable in the function and in the evaluation point, e.g. x = pi/4. The π, e, √, |x|, and ^ buttons insert symbols into whichever box you last clicked.
- Functions: sin, cos, tan, asin, acos, atan, sqrt (or √), abs — written abs(x) or with bars as |x| — ln (natural log), log (base 10), exp.
- Trigonometric functions work in radians.
Frequently asked questions
Does the calculator show the differentiation steps?
Yes. Every derivative is shown term by term with the rule used for each — power, product, quotient, chain, exponential, logarithm, and inverse trig rules. Evaluating at a point adds the substitution line with your value filled in.
Is the derivative exact or a numerical estimate?
Exact. The calculator differentiates the expression symbolically using the standard rules, not by finite differences, so f′(x) is an algebraic formula you can reuse. Evaluations at a point are computed from that formula and displayed in exact form (like 3/4 or π/2) when the value matches one.
Can it differentiate x^x or |x|?
Yes. x^x is handled with logarithmic differentiation, giving x^x·(ln(x) + 1). The absolute value |x| differentiates to x/|x|, which equals −1 for negative x and +1 for positive x; at x = 0 the derivative does not exist, and the calculator reports that when you evaluate there.
Why is my trig answer different from my textbook's?
Usually it is the same result written differently — for example d/dx tan(x) appears here as 1/cos(x)², which equals the textbook's sec²(x). The calculator uses only the functions it can parse, so secant and cosecant forms are expressed with cosine and sine.