Derivative Calculator

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Derivative

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What is Derivative?

The Derivative Calculator differentiates a polynomial with respect to one chosen letter. A derivative describes how a function changes as its input changes. In a graph, it gives the slope of the tangent line; in an applied problem, it can represent a rate such as velocity, marginal cost or growth per unit.

This page is intentionally limited to polynomials written with integer coefficients and non-negative whole-number powers. It handles expressions such as x^3 + 2x - 7. It does not parse parentheses, fractions, decimal coefficients, negative powers, roots, trigonometric functions or products of different variables. Keeping to the supported syntax makes the displayed symbolic result dependable.

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How to Use This Calculator

  1. Type the polynomial in Function f(x). Use a caret for a power, as in x^4, and place the coefficient directly before the variable, as in 3x.
  2. Enter one letter in Variable. The letter must match the one used in the polynomial; uppercase and lowercase are treated alike.
  3. Select Calculate. The calculator applies the power rule to every variable term, changes linear terms to their coefficients and removes constant terms.
  4. Read the result as another function. To find a slope at a particular input, substitute that input into the derivative afterward.

Spaces are optional, but explicit plus and minus signs must separate terms. Unsupported notation is rejected instead of being silently ignored.

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Formula Used

For a single power term, the power rule is d(ax^n)/dx = an x^(n−1). The coefficient a is multiplied by the old exponent n, then the exponent is reduced by one.

For a polynomial, differentiate term by term: d[f(x) + g(x)]/dx = f′(x) + g′(x). A constant has derivative zero, while a linear term ax has derivative a. These rules explain the entire calculation supported on this page.

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Variables Explained

Symbol or entryMeaning
Function f(x)The polynomial to differentiate, using integer coefficients and non-negative whole-number powers.
VariableThe single ASCII letter with respect to which the derivative is taken, such as x or t.
aThe signed numerical coefficient multiplying a variable term.
nThe original non-negative integer exponent in a term such as ax^n.
f′(x)The derivative function returned by the calculator.
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Worked Example

For f(x) = x^3 + 2x − 7, work through each term:

d(x^3)/dx = 3x^2
d(2x)/dx = 2
d(−7)/dx = 0

Combine the nonzero terms to obtain f′(x) = 3x^2 + 2. That is the exact text produced by the calculator. If you want the slope at x = 2, substitute after differentiating: f′(2) = 3(2^2) + 2 = 14.

As another check, the derivative of 5x^4 − 3x^2 + 9 is 20x^3 − 6x. The constant 9 disappears because changing x does not change a constant.

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Why Use This Calculator?

The output is a symbolic rate-of-change function, not automatically a slope at one point. Positive derivative values indicate that the original function is increasing locally; negative values indicate that it is decreasing locally. A derivative equal to zero identifies a stationary candidate, but further analysis is needed to decide whether it is a maximum, minimum or neither.

Units also change. If f measures distance in metres and x measures seconds, f′ has metres per second. If f is cost and x is quantity, f′ is cost per additional unit. The calculator performs algebra only, so the user must supply the appropriate interpretation and must check that a polynomial is a suitable model.

The derivative can itself be differentiated when the result remains a supported polynomial. A second derivative describes how the first rate changes and can help assess curvature. For f(x)=x^3+2x−7, the first derivative is 3x^2+2 and the second is 6x. This page performs one differentiation per submission, so enter the first result again if a later derivative is needed.

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Common Mistakes

    • Writing x*x or xx instead of the supported power notation x^2.
    • Entering a formula such as sin(x), 1/x or (x+1)^2 without expanding it into a supported polynomial first.
    • Using decimal or fractional coefficients even though this form currently accepts integer coefficients only.
    • Choosing a variable letter that differs from the letter in the expression.
    • Forgetting to multiply by the old exponent before reducing it.
    • Reading f′(x) as a numerical answer without substituting a particular x value.
    • Assuming every point where f′(x)=0 is automatically a maximum or minimum.
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Frequently Asked Questions

Can this calculator differentiate any function?

No. It supports a sum or difference of polynomial terms with integer coefficients and non-negative whole-number powers. Use a more general symbolic system for roots, fractions, products, trigonometric functions or logarithms.

Why does a constant disappear?

A constant does not change when the variable changes, so its rate of change is zero. Zero terms are omitted from the displayed polynomial.

Can I use a variable other than x?

Yes. Enter one letter, such as t, and use that same letter throughout the polynomial. The variable field does not accept a word or a regular-expression symbol.

How do I find the derivative at a point?

Calculate the derivative function first, then substitute the desired input into that result. For 3x^2+2 at x=2, the value is 14.

What is the derivative of a constant polynomial?

It is zero. For example, entering 12 produces Derivative = 0 because the function has the same value for every input.

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Sources & References

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