About the Derivative Calculator
This derivative calculator finds the derivative of a function of x symbolically — using the power, product, quotient and chain rules — and simplifies the result into a readable form. Type something like x^3 + 2x^2 − 5x + 1, sin(x)·e^x or ln(x^2 + 1) and it returns f′(x), and optionally the second or third derivative.
It then evaluates the derivative at the point you choose, giving the instantaneous rate of change (the slope of the curve there), the function value, and the equation of the tangent line y = mx + c. The graph plots the function, its derivative and the tangent line together so you can see where the curve rises, falls or turns.
It is built for calculus students checking homework, teachers preparing examples, and anyone in physics, economics or engineering who needs a rate of change such as velocity from position or marginal cost from a cost function. Use ln(x) for the natural logarithm; trigonometric functions work in radians.
How to use the derivative calculator
- 1Type the function using x as the variable, e.g. x^2*sin(x).
- 2Pick first, second or third derivative.
- 3Enter the x value where you want the slope.
- 4Read the derivative formula, its value and the tangent line equation.
- 5Check the graph to see where the derivative is positive, negative or zero.
Formula and method
The derivative is the limit of the slope of secant lines as the gap h shrinks to zero; it gives the instantaneous rate of change of f. In practice it is found with rules: the power rule d/dx xⁿ = n·xⁿ⁻¹, the product rule (uv)′ = u′v + uv′, the quotient rule (u/v)′ = (u′v − uv′)/v², and the chain rule (f(g(x)))′ = f′(g(x))·g′(x), plus known derivatives such as (sin x)′ = cos x, (eˣ)′ = eˣ and (ln x)′ = 1/x.
This calculator applies those rules symbolically, simplifies the result, and evaluates it at your point. The tangent line passes through (a, f(a)) with slope f′(a). Higher-order derivatives are found by differentiating repeatedly.
- f′(x)
- First derivative (slope / rate of change)
- f″(x)
- Second derivative (concavity, acceleration)
- a
- The x value where the derivative is evaluated
Worked examples
Polynomial at x = 2
d/dx (x³ + 2x² − 5x + 1) = 3x² + 4x − 5. At x = 2 that is 12 + 8 − 5 = 15, and f(2) = 7, so the tangent line is y = 15x − 23.
Second derivative of the same polynomial
Differentiating again gives f″(x) = 6x + 4, which is 16 at x = 2. It is positive, so the curve is concave up there.
sin(x) at 0
The derivative of sin x is cos x, and cos 0 = 1, so the tangent at the origin is y = x.
Product rule: x·eˣ at x = 1
By the product rule, (x eˣ)′ = eˣ + x eˣ = (x + 1)eˣ. At x = 1 that is 2e ≈ 5.4366, and f(1) = e ≈ 2.7183.
Square root at x = 4
By the power rule, d/dx x^(1/2) = ½x^(−1/2) = 1/(2√x). At x = 4 the slope is 1/(2 × 2) = 0.25 and f(4) = 2, so the tangent line is y = 0.25x + 1.
Natural log at x = 2
The derivative of ln x is 1/x, so the slope at x = 2 is 0.5, while ln 2 ≈ 0.6931.
Frequently asked questions
What does a derivative tell you?+
A derivative measures how fast a function is changing at a point — the slope of its tangent line. If position is a function of time, its derivative is velocity; if cost depends on quantity, the derivative is the marginal cost.
What is the power rule?+
The power rule says d/dx xⁿ = n·xⁿ⁻¹. For example, the derivative of x⁵ is 5x⁴, of √x = x^(1/2) is 1/(2√x), and of a constant is 0.
How do you find the equation of a tangent line?+
Evaluate f(a) and f′(a), then use y = f(a) + f′(a)(x − a). This calculator does this automatically and shows the result in slope-intercept form y = mx + c.
What does the second derivative mean?+
The second derivative is the rate of change of the slope. If f″ > 0 the graph is concave up (like a cup), if f″ < 0 it is concave down, and a sign change marks an inflection point.
Why does the answer look different from my textbook?+
A derivative can be written in many equivalent ways. The calculator rewrites common forms into textbook style, such as 1 / (2sqrt(x)) for the derivative of √x and 1 / (3x^(2 / 3)) for ∛x, but longer results may keep forms like sqrt(x)^2 instead of x. They are mathematically equal; the numeric value at your point confirms it.
What is the derivative of ln(x) and e^x?+
d/dx ln(x) = 1/x for x > 0, and d/dx eˣ = eˣ. With the chain rule, d/dx ln(g(x)) = g′(x)/g(x) and d/dx e^(g(x)) = g′(x)·e^(g(x)).