5
1
6
1
4
12
2
0.788675
0.211325
0
5
1
6
1
4
12
2
0.788675
0.211325
0
The First Derivative Calculator computes the first derivative of a cubic polynomial $$f(x) = ax^3 + bx^2 + cx + d$$ and evaluates it at any point. In addition to the derivative value, it finds all critical points (where the derivative equals zero), determines whether the function is increasing or decreasing at the evaluation point, and provides the tangent line equation. This is a focused, precision tool for polynomial differentiation and critical point analysis.
The first derivative of a function is the cornerstone of differential calculus. For a polynomial, the differentiation process is governed by the power rule, which states that $$\frac{d}{dx}[x^n] = nx^{n-1}$$. Applying this term by term to $$f(x) = ax^3 + bx^2 + cx + d$$ yields:
$$f'(x) = 3ax^2 + 2bx + c$$
This derivative is itself a quadratic polynomial, and finding its roots gives the critical points of the original cubic — the x-values where the function changes from increasing to decreasing or vice versa. These critical points are candidates for local maxima and local minima.
The critical points of $$f'(x) = 3ax^2 + 2bx + c$$ are found using the quadratic formula:
$$x = \frac{-2b \pm \sqrt{4b^2 - 12ac}}{6a}$$
The discriminant $$\Delta = 4b^2 - 12ac$$ determines the nature of the critical points. If $$\Delta > 0$$, there are two distinct critical points and the cubic has both a local maximum and a local minimum. If $$\Delta = 0$$, there is exactly one critical point (an inflection point with horizontal tangent). If $$\Delta < 0$$, the cubic has no critical points and is strictly monotonic — always increasing if $$a > 0$$ or always decreasing if $$a < 0$$.
The sign of $$f'(x)$$ at any point tells you the function's behavior there. When $$f'(x) > 0$$, the function is increasing — as x grows, f(x) grows. When $$f'(x) < 0$$, the function is decreasing. When $$f'(x) = 0$$, the function has a horizontal tangent — a critical point that could be a local extremum or an inflection point.
Applications of first derivative analysis include optimization problems in engineering and economics (finding maximum profit or minimum cost), curve sketching in mathematics, velocity analysis in physics (where position is a cubic function of time), and root finding in numerical methods where Newton's method uses the first derivative iteratively. The tangent line at any point gives the best linear approximation, used extensively in linearization and Taylor series.
Enter the four coefficients of your cubic polynomial and the evaluation point to see the complete first-derivative analysis.
The calculator applies the power rule to differentiate the cubic polynomial $$f(x) = ax^3 + bx^2 + cx + d$$.
Step 1: Differentiate.
$$f'(x) = 3ax^2 + 2bx + c$$
Step 2: Evaluate at the given point. Substitute the x-value into both $$f(x)$$ and $$f'(x)$$.
Step 3: Find critical points. Solve $$f'(x) = 0$$, i.e., $$3ax^2 + 2bx + c = 0$$, using the quadratic formula:
$$x_{1,2} = \frac{-2b \pm \sqrt{(2b)^2 - 4(3a)(c)}}{2(3a)} = \frac{-2b \pm \sqrt{4b^2 - 12ac}}{6a}$$
Step 4: Determine behavior. If $$f'(x) > 0$$ at the evaluation point, the function is increasing there. If $$f'(x) < 0$$, it is decreasing. If $$f'(x) = 0$$, it is at a critical point.
Step 5: Tangent line. The tangent line at $$(x_0, f(x_0))$$ is:
$$y = f'(x_0) \cdot x + \left[f(x_0) - f'(x_0) \cdot x_0\right]$$
f(x) is the value of the cubic polynomial at the evaluation point.
f'(x) is the slope of the function at that point — the instantaneous rate of change.
Function Behavior indicates whether the function is increasing, decreasing, or stationary at the point. This is determined solely by the sign of f'(x).
Critical Points x₁ and x₂ are the x-values where the derivative equals zero. If the discriminant is negative, no real critical points exist and both values display NaN. A single critical point occurs when the discriminant is exactly zero.
Discriminant of the quadratic f'(x) = 3ax² + 2bx + c. Positive means two critical points; zero means one; negative means none.
Tangent Line Slope and y-intercept define the tangent line y = mx + B at the evaluation point, where m = f'(x₀) and B = f(x₀) − f'(x₀)·x₀.
Inputs
Results
f'(x) = 6x² − 6x. Setting f'(x)=0: 6x(x−1)=0, so x=0 and x=1. At x=0.5: f'(0.5) = 6(0.25) − 6(0.5) = 1.5 − 3 = −1.5 < 0, so the function is decreasing between the two critical points.
Inputs
Results
f'(x) = 3x² + 3. Since 3x² + 3 > 0 for all real x, the function is always increasing. Discriminant = 0 − 12(1)(3) = −36 < 0, confirming no real critical points. At x=2: f'(2) = 12 + 3 = 15.
The first derivative tells you the rate of change of a function. It determines where the function is increasing (f' > 0), decreasing (f' < 0), or stationary (f' = 0). In physics, if f(x) represents position, f'(x) is velocity. In economics, if f(x) is total cost, f'(x) is marginal cost.
Critical points are x-values where f'(x) = 0 or f'(x) is undefined. For polynomials, only f'(x) = 0 applies. Critical points are important because they are the only locations where local maxima or local minima can occur (by Fermat's theorem). Finding critical points is the first step in optimization problems.
A negative discriminant means the quadratic equation f'(x) = 0 has no real solutions. For the original cubic, this means there are no critical points — the function is strictly monotonic (always increasing or always decreasing). This occurs when the cubic's curve has no local peaks or valleys.
Use the second derivative test: compute f''(x) at the critical point. If f''(x) > 0, it is a local minimum (concave up). If f''(x) < 0, it is a local maximum (concave down). For the cubic f(x) = ax³ + bx² + cx + d, f''(x) = 6ax + 2b. Alternatively, check the sign of f'(x) on either side of the critical point.
This calculator is designed for cubic polynomials (degree 3 or lower). Set a = 0 for quadratics, or a = b = 0 for linear functions. For higher-degree polynomials, the derivative formulas require additional terms and the critical point analysis becomes more complex (cubic or higher equations).
The tangent line is the best linear approximation of the function near a given point. It is used in linearization, Newton's method for root finding, and differential approximation where f(x + Δx) ≈ f(x) + f'(x)·Δx. The tangent line captures the local behavior of the function at first order.
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