d/dx f(x) -- step by step
Type a function -- the formula renders live. Press "=" and the calculator builds the derivative along with a full chain of reasoning: which rule applies at each step and why.
Like any software, the calculator can contain bugs. For exams, publications or other important work, verify the result independently.
The online derivative calculator finds the derivative of a function symbolically and shows a full step-by-step solution: which rule applies at each stage of the computation and why. Derivatives of order 1 through 5 are supported for ordinary functions, and there's a separate mode for implicit differentiation when y isn't isolated on one side of the equation.
The service covers the standard differentiation toolkit: the sum rule, the product rule, the quotient rule, the chain rule, the power rule, and table derivatives of polynomial, trigonometric, hyperbolic, exponential, and logarithmic functions. When both the base and the exponent of a power depend on the variable (as in x^x), the calculator switches to logarithmic differentiation and explains why. Each rule step shows both the general formula and the reduced result on the same line -- d/dx[x^2] = 2x^(2-1) = 2x, for example -- so nothing is left half-simplified.
The graph shows f(x) alongside f'(x), and any higher derivative you compute is added to it too via "Calculate next higher derivative." Typing a point shows the tangent line there, and an optional overlay shades the intervals where the function is increasing or decreasing, based on the sign of the derivative -- with local extrema marked where the derivative genuinely changes sign (a point where it merely touches zero, like x = 0 for x²sin(x), is correctly left unmarked) and inflection points marked where the concavity actually flips. The chart supports panning and zooming, resampling the function as needed rather than stretching a fixed set of points.
Beyond the closed-form derivative, the "Roots/zeros" button finds every exact zero of the derivative (solved symbolically where possible, with a numeric fallback noted when it isn't), and "Simplify" looks for an equivalent, more compact form of the result. A separate panel finds the absolute maximum and minimum of the function on a specific closed interval [a, b] -- checking the interval's endpoints together with every critical point strictly inside it, per the Extreme Value Theorem -- and flags the result if the function is discontinuous or undefined somewhere on that interval, or if a critical point could only be pinned down numerically.
The calculator also includes an answer-checking feature: submit the derivative you computed by hand, and it's compared against the correct one symbolically, accepting any mathematically equivalent form rather than requiring one specific representation.
Switching to "Implicit: F(x, y) = 0" mode handles equations that relate x and y without y ever being isolated -- the circle x² + y² = 25, for instance, or anything built the same way (an ellipse, x·y + sin(y) = x, and so on). Enter the equation, pick which variable to differentiate with respect to, and the calculator treats the other one as an implicit function of it, differentiates both sides, and solves for dy/dx algebraically. The step-by-step breakdown uses the exact same rule set as ordinary differentiation -- product rule, chain rule, and so on -- since every occurrence of y just contributes a factor of y' via the chain rule as the equation is walked term by term.
The mathematical expression is first parsed syntactically. The calculator recognizes implicit multiplication (for example, 2sin(x) is interpreted as 2*sin(x)) and function aliases (tg, ctg, sh, and ch are recognized alongside tan, cot, sinh, and cosh).
Differentiation itself is a recursive descent over the expression tree: at each node the calculator recognizes whether it's looking at a sum, a product, a quotient, a power, or an elementary function applied to something, and applies the corresponding rule -- recursing into the non-trivial parts until every piece has been reduced to a table derivative. This is a fundamentally more constrained problem than symbolic integration, so no search over alternative solution paths is needed.
For higher orders, the same process repeats: the first derivative is computed and explained in full, then differentiated again to get the second, and so on up to the fifth.
Enter the function you want to differentiate without f(x) =; it doesn't need to be included. The formula is rendered graphically as you type, so you can immediately check that the calculator has interpreted the expression as intended.
Use parentheses wherever they are needed to specify the order of operations, for example a/(b+c). Use a period rather than a comma for decimal numbers: 3.141, not 3,141. The examples section includes the most commonly entered functions and can be used as a reference for the syntax.
Choose the derivative order (1st through 5th) from the dropdown before pressing "=". Once the expression has been entered, click "=" and the calculator will display the derivative, the step-by-step solution, and the graph.
For an equation relating x and y, switch to the "Implicit" mode above the input field first -- the equation field replaces the single-function one, and "differentiate w.r.t." offers whichever variables it detects in what you've typed so far, defaulting to x.