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Derivative Calculator

Calculate the derivative of any cubic polynomial f(x) = ax^3 + bx^2 + cx + d using the power rule, and evaluate it at any point x. The calculator returns both the derivative polynomial f'(x) = 3ax^2 + 2bx + c and its numerical value at your chosen x, which represents the slope of the tangent line at that point.

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Formula

f'(x) = 3ax^2 + 2bx + c for f(x) = ax^3 + bx^2 + cx + d.

The power rule states that the derivative of x^n is n times x^(n-1). Applied term by term to a cubic: the derivative of ax^3 is 3ax^2, bx^2 becomes 2bx, cx becomes c, and the constant d becomes 0. The calculator then substitutes your x value into the derivative polynomial to find the slope at that specific point.

Worked Example

f(x) = 2x^3 - 3x^2 + x - 5 evaluated at x = 2: Step 1: Apply power rule: f'(x) = 6x^2 - 6x + 1 Step 2: Substitute x = 2: f'(2) = 6(4) - 6(2) + 1 = 24 - 12 + 1 = 13 Step 3: The slope of the tangent line at x = 2 is 13 Step 4: Original function value: f(2) = 16 - 12 + 2 - 5 = 1 The tangent line at (2, 1) has slope 13.

Frequently Asked Questions

Why is this limited to cubic (degree 3) polynomials?

This calculator covers polynomials up to degree 3 (ax^3 + bx^2 + cx + d) with a clean interface. For higher-degree polynomials, set unused coefficients to 0. For quadratics, set a = 0. For linear functions, set a = 0 and b = 0.

What does the derivative value at a point mean?

The derivative f'(x) at a specific x gives the instantaneous rate of change, which is the slope of the tangent line to the curve at that point. A positive value means the function is increasing, negative means decreasing, and zero means a potential local maximum or minimum.

Can I use this for exponential or trigonometric functions?

No. This calculator uses the polynomial power rule only. For derivatives of sin, cos, e^x, ln(x), and other functions, use a symbolic calculator or computer algebra system.

Can coefficients be decimals or negative numbers?

Yes. All coefficients accept any real number including decimals and negatives. The power rule applies identically regardless of coefficient sign or magnitude.

How do I find where the derivative equals zero?

Set the derivative polynomial 3ax^2 + 2bx + c = 0 and solve the quadratic. The solutions are critical points where the original function may have local maxima, minima, or inflection points. This calculator evaluates f'(x) at a single point; solving for zeros requires the quadratic formula.

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