Differentiate a polynomial-style expression in x with a derivative calculator that shows steps, first and second derivatives, slope at a point, concavity.
Last updated
Examples
Supported input scope
This derivative calculator is for single-variable polynomial-style expressions built from constants and powers of x, including negative and fractional exponents such as x^-2 or x^0.5.
It does not parse products like (x+1)(x-1), quotients,
trig functions, exponentials, or nested chain-rule expressions.
Derivative result
6x^2 + 6x - 5
Derivative of 2x^3 + 3x^2 - 5x + 4 using the power rule term by term.
Original function
2x^3 + 3x^2 - 5x + 4
Derivative
6x^2 + 6x - 5
Second derivative
12x + 6
Point evaluation
f'(2) = 31
At x = 2, the original function value is 22, f''(2) = 30, and the tangent line is y = 31x - 40.
Local behavior at the chosen point
At x = 2, the function is increasing; the second derivative is 30, so the curve is concave up at that point.
Step-by-step rule application
Each term is differentiated independently, then the results are combined into the final derivative.
Step 1: 2x^3 → 6x^2
Apply the power rule: multiply 2 by 3, then reduce the exponent by 1.
Step 2: 3x^2 → 6x
Apply the power rule: multiply 3 by 2, then reduce the exponent by 1.
Step 3: -5x → -5
Apply the power rule: multiply -5 by 1, then reduce the exponent by 1.
Derivative calculator guide: polynomial derivatives, slope at a point
A derivative calculator helps you turn a polynomial expression into its derivative, then use that derivative to answer practical calculus questions such as “what is the slope at x = a?” and “what is the tangent line there?”.
What this derivative calculator is actually solving
This calculator differentiates a sum of powers of x term by term. In other words, it is built for expressions such as 2x^3 + 3x^2 − 5x + 4, 4x^0.5 − 3x + 9, or 5x^-2 − 7x + 1. Those are all single-variable polynomial-style expressions in which each term can be read as a coefficient multiplied by x raised to a real exponent.
That scope matters because many people search for “derivative calculator” expecting a full computer algebra system. This page is not trying to be that. It does not expand products, simplify quotients, or symbolically differentiate functions like sin(x), ln(x), or e^x. Instead, it gives a cleaner experience for the common classroom and homework workflow where the power rule is the right tool and the next question is usually the slope or tangent line at a chosen x-value.
That makes the page especially useful for algebra-to-calculus transitions. If you already know the expression is a sum of powers of x, the hard part is usually not the symbolic manipulation itself but keeping the notation straight, seeing how each term changes, and connecting the derivative to slope rather than treating it as a purely mechanical output.
The power rule and why it works term by term
For a term ax^n, the derivative is n·a·x^(n−1). Constants have derivative zero, and the derivative of a sum is the sum of the derivatives. That is why polynomial-style expressions are a natural fit for a calculator like this one: each term can be handled independently and then recombined into one final derivative.
The power rule still works when exponents are negative or fractional. For example, the derivative of x^0.5 is 0.5x^-0.5, and the derivative of x^-2 is -2x^-3. The symbolic pattern is the same as it is for positive whole-number exponents. What changes is the shape of the resulting exponent and how you interpret it near points where the original function may be undefined or steep.
This is also where users often confuse “power rule” with “all derivative rules”. The power rule is enough for sums of powers of x, but it is not enough on its own for products like (x+1)(x-1), quotients, compositions such as (3x+1)^5 written as a chain-rule problem, or trig and logarithmic functions. Those need broader symbolic rules than this calculator currently supports.
d/dx(axⁿ) = n·a·x^(n−1)
Power rule for a single term with coefficient a and exponent n.
d/dx(c) = 0
Constants do not change with x, so their derivative is zero.
d/dx[f(x) + g(x)] = f′(x) + g′(x)
The derivative of a sum can be found one term at a time and then recombined.
Worked example: derivative, slope, and tangent line
Take f(x) = 2x^3 + 3x^2 − 5x + 4. Applying the power rule term by term gives f′(x) = 6x^2 + 6x − 5. That symbolic answer is already useful, but many students and working users actually need the next step more: evaluating the derivative at a point.
At x = 2, the derivative is f′(2) = 6(2^2) + 6(2) − 5 = 24 + 12 − 5 = 31. That means the slope of the tangent line to the curve at x = 2 is 31. The original function value is f(2) = 22, so the tangent line through the point (2, 22) with slope 31 is y = 31x − 40.
This is why the point-evaluation panel matters. A derivative is not only a new algebraic expression; it is also a slope function. Once you choose a specific x-value, the derivative tells you the instantaneous rate of change there, and that lets you build a tangent line or compare how steep the function is at different points.
The derivative tells you how fast the original function is changing at each x-value. When the derivative is positive, the function is increasing at that point. When the derivative is negative, the function is decreasing there. When the derivative is zero, the graph may flatten temporarily, which is why derivatives are often used to study turning points and critical points.
That interpretation becomes much more concrete when you evaluate the derivative at a specific x-value. A derivative value of 31 means the function is increasing very steeply there. A derivative value of -2 means the function is falling gently. A derivative value close to zero means the graph is relatively flat at that location. The units of the derivative are also worth remembering: they are “output units per unit of x”.
For classroom work, this connects differentiation to graphing. For applied use, it connects differentiation to rates such as speed, marginal change, and local sensitivity. Even on a polynomial-only page, that conceptual bridge matters because it keeps the calculator from being just a symbolic answer box.
Second derivative and local behavior
Many derivative calculator searches are really looking for more than the first derivative. After f′(x) tells you the local slope, f″(x) tells you how that slope is changing. This page now shows the second derivative for the same polynomial-style expression, so you can connect a first derivative answer to concavity and local behavior without leaving the power-rule workflow.
When you enter an x-value, the calculator evaluates f(x), f′(x), and f″(x) together. A positive first derivative means the function is increasing at that point, a negative first derivative means it is decreasing, and a zero first derivative means the graph is locally flat. A positive second derivative indicates concave-up behavior, while a negative second derivative indicates concave-down behavior.
This is not the same as a full critical-point solver, but it is a practical check for homework and graph interpretation. It helps you see whether the tangent line, slope at a point, and second derivative tell a coherent story before you move on to a graphing calculator or a broader symbolic derivative solver.
f″(x) = d/dx[f′(x)]
The second derivative is the derivative of the first derivative.
f′(a) > 0 means increasing; f′(a) < 0 means decreasing
Point-evaluation interpretation for the first derivative.
f″(a) > 0 means concave up; f″(a) < 0 means concave down
Point-evaluation interpretation for the second derivative.
What this calculator does not cover
This derivative calculator is deliberately narrower than tools like Wolfram or Symbolab. It does not parse trig functions, logarithms, exponentials, products, quotients, or nested expressions that need chain, product, or quotient rules. If you enter sin(x), (x+1)(x-1), or (x^2+1)/(x-3), the page should be treated as out of scope rather than as a general symbolic engine.
It also assumes the expression is already written as a sum of powers of x. That means it does not expand brackets for you, simplify algebraic identities, or detect alternative but equivalent forms. If your expression needs simplification first, do that before using this page, or move to a broader CAS-style derivative solver.
Those limitations are not defects so much as scope boundaries. A narrow, explicit scope is better than a misleading promise. The page is strongest when you want fast, readable power-rule differentiation with a clean slope-at-a-point interpretation, not when you want all of calculus symbolics in one place.
Further reading
OpenStax Calculus Volume 1 — Derivatives — OpenStax calculus reference covering derivative concepts, notation, and the transition from average to instantaneous rate of change.
The derivative of any constant is zero because constants do not change with respect to x.
Does this work for negative or fractional exponents?
Yes. The power rule still applies to real exponents, including negative and fractional powers such as x^-2 or x^0.5. The derivative may introduce negative exponents or expressions that are undefined at some x-values, but the symbolic differentiation rule is the same.
Can this derivative calculator handle sin(x), ln(x), or e^x?
No. This page is limited to polynomial-style expressions made from sums of powers of x. Trigonometric, logarithmic, exponential, product-rule, quotient-rule, and chain-rule expressions are outside scope here.
What does f′(a) mean?
f′(a) means the derivative evaluated at x = a. It gives the slope of the tangent line to the original function at that point, and it represents the instantaneous rate of change there.
Why does the derivative tell me the slope?
Because the derivative is defined from the limit of secant slopes as two points on the graph move together. That limiting slope is the tangent-line slope, so evaluating the derivative at a point tells you how steep the graph is at that exact location.
What if the derivative equals zero?
A derivative of zero means the graph is locally flat at that point. That can indicate a local maximum, local minimum, or a flat point that is not a turning point. The derivative alone shows the slope there, but not the full graph behavior on its own.
Can this tool find the tangent-line equation?
Yes, if you enter an x-value in the point-evaluation field. The calculator evaluates both the original function and its derivative at that point, then uses the point and slope to build the tangent-line equation.
Does this derivative calculator show the second derivative?
Yes. For supported polynomial-style inputs, it shows f′(x) and f″(x). If you enter an x-value, it also evaluates the second derivative there so you can interpret concavity at the same point as the tangent-line slope.
Can I type 2*x^3 instead of 2x^3?
Yes. The calculator accepts an explicit multiplication mark between a number and x, such as 2*x^3. It still does not expand products like (x+1)(x-1), so factored expressions need to be rewritten as sums of powers first.
Why does the page reject (x+1)(x-1) even though it is a polynomial?
Because the page expects the input to be written as a sum of powers of x, not as a factored product. (x+1)(x-1) is a polynomial after expansion, but this calculator does not expand brackets for you. You would need to rewrite it first as x^2 - 1.
Does the calculator combine like terms automatically?
Yes. If the parsed input includes terms with the same exponent, the calculator combines them before building the final expression and derivative. That keeps the displayed result cleaner and makes the step list easier to read.
Is this the same as a full symbolic derivative solver?
No. A full symbolic solver can apply many derivative rules across a much wider expression grammar. This page is intentionally narrower: it focuses on power-rule differentiation for sums of powers of x, plus slope-at-a-point interpretation.
Guides
Featured in articles
Step-by-step guides that use this calculator to solve real problems.
