On this page
How to use the derivative calculator
- Type a function in the Function or equation box, for example
x^2 sin(x), then press Enter or Differentiate. - Pick an Order from 1st to 5th if you need f'', f''' and beyond. Each order gets its own collapsible list of steps.
- If the function has more than one letter, choose the Variable to differentiate by. Every other letter is held constant, which makes this a partial derivative calculator.
- Optionally type a number in At point. You get f and f' at that value and, for a first derivative, the tangent line.
- Read the Steps from top to bottom. Each line names a rule, shows the piece it was applied to, and shows what that piece became.
The graph underneath draws f in blue and the derivative in red. With a point entered it adds the tangent line in green. Wherever the red curve crosses zero, the blue curve has a flat tangent.
| You mean | Type |
|---|---|
| Powers and roots | x^3, x^(1/3), sqrt(x), cbrt(x) |
| Multiplication | 2x, x^2 sin(x), x*y |
| Trig and inverse trig | sin(x), tan(x), atan(2x), asin(x) |
| Exponentials | e^(2x), exp(x), 2^x |
| Logarithms | ln(x) is natural, log(x) is base 10 |
| Absolute value | abs(x) or |x| |
| Implicit differentiation | an equation, such as x^2 + y^2 = 25 |
Always put a function's argument in parentheses. 3sin x is not read the way you mean it, while 3 sin(x) is.
The math behind it: the rules at a glance
The derivative of f at a point is the slope of its tangent line there. Formally it is the limit of (f(x+h) - f(x)) / h as h shrinks to 0. Nobody takes that limit by hand for every problem. Instead you split the function into pieces and apply a rule to each one. The calculator does the same and lists each rule it used, so you can see which rule applies where.
| Rule | Pattern | Example you can type | Result |
|---|---|---|---|
| Constant | d/dx c = 0 | 7 | 0 |
| Power | d/dx xⁿ = n·xⁿ⁻¹ | x^5 | 5x⁴ |
| Constant multiple and sum | differentiate each term, keep the coefficients | 3x^4 - 2x^2 + 7 | 12x³ - 4x |
| Product | (uv)' = u'v + uv' | x^2 sin(x) | 2x·sin(x) + x²·cos(x) |
| Quotient | (u/v)' = (u'v - uv') / v² | sin(x)/x | (cos(x) - sin(x)/x)/x |
| Chain | derivative of the outside times derivative of the inside | sin(x^2) | 2x·cos(x²) |
| Exponential | d/dx e^u = e^u·u'; d/dx a^u = a^u·ln(a)·u' | e^(3x), 2^x | 3e^(3x); 2^x·ln 2 |
| Logarithm | d/dx ln(x) = 1/x | ln(x), log(x) | 1/x; 1/(x·ln 10) |
| Trig | standard table | cos(x), tan(x) | -sin(x); sec(x)² |
| Variable in the exponent | logarithmic differentiation | x^x | x^x·(ln(x) + 1) |
Two extensions use the same machinery. A partial derivative differentiates by one variable and treats the others as constants. An implicit derivative handles an equation F(x, y) = 0 where y is not isolated: the calculator moves everything to one side, finds the partial derivatives Fx and Fy, and returns dy/dx = -Fx / Fy. Before a result is shown, it is checked numerically against the derivative built into math.js, so a slip in a rule gets caught.
Worked examples with every step
1. Product rule: x² sin(x)
Here u = x² and v = sin(x). The calculator lists three steps:
- Power rule: d/dx [x²] = 2x
- Derivative of sine: d/dx [sin(x)] = cos(x)
- Product rule: d/dx [x²·sin(x)] = 2x·sin(x) + x²·cos(x)
At x = 1 the tool gives f(1) = 0.8414709848 and f'(1) = 2.223244275. You can confirm by hand: 2·sin(1) + cos(1) = 1.6829 + 0.5403 = 2.2232. Setting the order to 2 adds seven steps and gives f''(x) = 2(2x·cos(x) + sin(x)) - x²·sin(x), which equals 3.002680208 at x = 1.
2. Chain rule: ln(x² + 1)
The outside function is ln, the inside is x² + 1. The steps are the power rule for x² (giving 2x), the sum rule for x² + 1 (giving 2x), and the chain rule for the natural log, which divides the inside's derivative by the inside:
d/dx [ln(x² + 1)] = 2x/(x² + 1)
At x = 2 that is f(2) = 1.609437912 (this is ln 5) and f'(2) = 0.8, which matches 2·2 / (4 + 1) = 4/5.
3. Quotient rule and a tangent line: e^(2x) / x at x = 1
Steps: constant multiple rule for 2x (giving 2), the exponential rule (2e^(2x)), then the quotient rule, which returns e^(2x)·(2/x - (1/x)²). With the point set to 1, the tool reports f(1) = 7.389056099 and f'(1) = 7.389056099. Both are e², since 2/1 - 1 = 1. The tangent line it prints is y = 7.3890561x + 0. A zero intercept means this particular tangent passes through the origin, an easy visual check on the green line in the graph.
4. Implicit differentiation: x² + y² = 25
This circle has no single y = f(x) form, so the calculator works with F(x, y) = x² + y² - 25. It shows Fx = 2x and Fy = 2y, then dy/dx = -(x/y). The point box is not used in this mode, so you substitute yourself: at (3, 4) the slope is -3/4 = -0.75, and at (3, -4) it is +0.75. The tangent at (3, 4) is y - 4 = -0.75(x - 3), or y = -0.75x + 6.25. If you want a tangent line calculator for an explicit function, use examples 1 to 3 instead.
Reading the result and common mistakes
Why a correct answer can look different
Two expressions can be the same function written differently. For sin(x)/x the textbook answer is (x·cos(x) - sin(x)) / x². The calculator shows (cos(x) - sin(x)/x) / x. Compare them numerically: at x = 1 both equal -0.3011686789, and at x = 2 both equal -0.4353977750. Likewise, e^(2x)·(2/x - (1/x)²) is the same as e^(2x)·(2x - 1)/x². If your homework answer disagrees in form, plug in two or three values of x. Matching numbers mean you are both right.
log versus ln
Here log means base 10 and ln means natural log, the same convention as most scientific calculators. This changes the answer: d/dx ln(x) = 1/x, but d/dx log(x) = 1/(x·ln 10). For log(x² + 1) at x = 1 the slope is 0.4342944819, not 1. Many textbooks write log for the natural log, so check which one your course uses before typing.
Other things to watch
- Radians. Trig derivatives such as d/dx sin(x) = cos(x) hold only when x is in radians, and the value at a point is computed in radians. See degrees versus radians if your problem is in degrees.
- Forgetting the inside. d/dx sin(x²) is 2x·cos(x²), not cos(x²). A chain rule step in the list is your reminder.
- Undefined points. |x| has no derivative at 0 and 1/x has none at 0, so a point there gives no usable slope.
- Orders. Higher orders differentiate the simplified previous result, so the step lists get longer quickly.
To see what a derivative means numerically, try the numerical differentiation guide, or use the scientific calculator for the arithmetic around your answer.
Frequently asked questions
Is the derivative calculator with steps really free?
Yes. Every rule used is listed as a step, for every order, with no paywall in front of the steps. You can also differentiate by any variable, enter a point and get the tangent line.
Why does my textbook show a different answer?
Usually both are correct and only the form differs, for example a combined fraction versus a nested one. Evaluate both at two or three values of x. If the numbers match, the answers are equivalent.
How do I use it as an implicit differentiation calculator?
Type the whole equation, such as x^2 + y^2 = 25. It returns dy/dx = -Fx / Fy and shows Fx and Fy. Substitute your point into the formula yourself.
What is the difference between a partial and an implicit derivative?
A partial derivative differentiates a function of several variables with respect to one of them, treating the rest as constants. An implicit derivative finds dy/dx from an equation where y is not solved for, using dy/dx = -Fx / Fy.
Does log mean base 10 or natural log?
log is base 10 and ln is natural. The derivative of log(x) is 1/(x·ln 10), while the derivative of ln(x) is 1/x.
How do I find a tangent line, and how many derivatives can I take?
Enter a value in the At point box with the order set to 1st. You get f(a), f'(a) and the line y = f'(a)·x + b, drawn in green on the graph. You can take derivatives of order 1 to 5.