3
7
10
81.8699
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-4
-0.142857
1
3
7
10
81.8699
deg
-4
-0.142857
1
The Derivative Calculator computes the first and second derivatives of common function types evaluated at any point. The derivative measures the instantaneous rate of change of a function — it is the fundamental operation of differential calculus. This calculator supports polynomial, trigonometric (sine and cosine), exponential, and logarithmic functions, providing the function value, first derivative, second derivative, tangent line angle, and tangent line y-intercept at your chosen point.
The derivative concept emerged from two independent discoveries in the 17th century. Isaac Newton developed his method of "fluxions" to describe instantaneous velocity and acceleration in physics. Gottfried Wilhelm Leibniz, working independently, created the $$\frac{dy}{dx}$$ notation that we still use today. Their combined insights revealed that rates of change and accumulation (integration) are inverse operations — the Fundamental Theorem of Calculus, which unifies differential and integral calculus.
Formally, the derivative of $$f(x)$$ at a point $$x$$ is defined as:
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
Rather than computing this limit numerically, this calculator uses exact symbolic derivative formulas for each function type. For a polynomial $$f(x) = ax^3 + bx^2 + cx + d$$, the power rule gives $$f'(x) = 3ax^2 + 2bx + c$$ exactly. For trigonometric functions, the chain rule yields $$\frac{d}{dx}[a\sin(bx+c)] = ab\cos(bx+c)$$. For exponential functions, $$\frac{d}{dx}[ae^{bx}] = abe^{bx}$$. For logarithmic functions, $$\frac{d}{dx}[a\ln(bx)] = \frac{a}{x}$$.
The second derivative $$f''(x)$$ measures the rate of change of the rate of change — geometrically, it describes the concavity of the curve. A positive second derivative means the curve is concave up (bowl-shaped), while a negative second derivative means concave down. Points where $$f''(x) = 0$$ are potential inflection points where the concavity changes.
The tangent line at a point $$(x_0, f(x_0))$$ has slope $$f'(x_0)$$ and equation $$y = f'(x_0)(x - x_0) + f(x_0)$$. The tangent line is the best linear approximation of the function near that point. The angle of the tangent line with the horizontal axis is $$\theta = \arctan(f'(x_0))$$, which this calculator reports in degrees.
Derivatives are ubiquitous in science and engineering. In physics, velocity is the derivative of position and acceleration is the derivative of velocity. In economics, marginal cost is the derivative of total cost. In optimization, setting the derivative to zero finds maxima and minima. In machine learning, gradient descent uses derivatives to minimize loss functions. Select your function type, enter the coefficients and evaluation point, and explore the derivative's behavior.
The calculator applies exact differentiation rules to compute $$f(x)$$, $$f'(x)$$, and $$f''(x)$$ at the specified point.
Polynomial: $$f(x) = ax^3 + bx^2 + cx + d$$
$$f'(x) = 3ax^2 + 2bx + c$$
$$f''(x) = 6ax + 2b$$
Sine: $$f(x) = a\sin(bx + c)$$
$$f'(x) = ab\cos(bx + c)$$
$$f''(x) = -ab^2\sin(bx + c)$$
Cosine: $$f(x) = a\cos(bx + c)$$
$$f'(x) = -ab\sin(bx + c)$$
$$f''(x) = -ab^2\cos(bx + c)$$
Exponential: $$f(x) = ae^{bx}$$
$$f'(x) = abe^{bx}$$
$$f''(x) = ab^2 e^{bx}$$
Logarithmic: $$f(x) = a\ln(bx)$$ for $$bx > 0$$
$$f'(x) = \frac{a}{x}$$
$$f''(x) = -\frac{a}{x^2}$$
Tangent line: The tangent at $$(x_0, f(x_0))$$ is $$y = f'(x_0) \cdot x + B$$ where $$B = f(x_0) - f'(x_0) \cdot x_0$$.
Angle: $$\theta = \arctan(f'(x_0)) \times \frac{180}{\pi}$$ degrees.
f(x) is the function value at the evaluation point. This is the y-coordinate of the curve at that x-value.
f'(x) is the first derivative — the instantaneous rate of change and the slope of the tangent line. Positive values mean the function is increasing; negative values mean it is decreasing; zero indicates a potential maximum, minimum, or inflection point.
f''(x) is the second derivative — the rate of change of the slope. Positive means concave up (the curve bends upward, like a bowl). Negative means concave down (the curve bends downward, like an arch). Zero at a point where the sign changes indicates an inflection point.
Tangent Line Angle gives the angle in degrees that the tangent line makes with the positive x-axis. A 0° angle means the function is momentarily flat; 45° means slope = 1; 90° approaches a vertical tangent.
Tangent Line y-intercept gives B in the tangent line equation y = f'(x₀)·x + B. Together with f'(x₀), this fully defines the tangent line.
Inputs
Results
f(1) = 1 + 2 = 3. f'(x) = 3x² + 4x, so f'(1) = 3 + 4 = 7. f''(x) = 6x + 4, so f''(1) = 10 > 0 (concave up). Tangent line: y = 7x − 4. Angle = arctan(7) ≈ 81.87°.
Inputs
Results
f(0.5) = 2sin(π/2) = 2. f'(x) = 2π·cos(πx), f'(0.5) = 2π·cos(π/2) ≈ 0. The function is at a maximum — the tangent is horizontal. f''(0.5) = −2π²·sin(π/2) ≈ −19.74 < 0 confirms concave down (maximum).
The derivative $$f'(x)$$ is a function that gives the rate of change at each point. The differential $$df = f'(x)\,dx$$ is an infinitesimal change in the function value corresponding to an infinitesimal change in the input. The derivative is a ratio; the differential is the numerator of that ratio when written as $$df/dx$$.
Symbolic differentiation provides exact results with no approximation error. Numerical differentiation (using finite differences) introduces truncation and round-off errors. For the supported function types, the derivative formulas are well-known and can be applied exactly, making symbolic computation the superior approach.
A zero derivative at a point means the function is momentarily neither increasing nor decreasing — the tangent line is horizontal. This occurs at local maxima, local minima, and saddle points (inflection points with horizontal tangent). The second derivative test helps distinguish: f''(x) > 0 indicates a minimum, f''(x) < 0 indicates a maximum, and f''(x) = 0 is inconclusive.
The tangent line angle is the angle between the tangent line and the positive x-axis. At 0° the function is flat, at 45° the slope is 1 (rising at a 1:1 ratio), and angles approaching ±90° indicate extremely steep slopes. Negative angles mean the function is decreasing.
This calculator handles the most common function families: polynomials (up to degree 3), sine, cosine, exponential, and natural logarithm. For products, quotients, or compositions of these, you would apply the product rule, quotient rule, or chain rule manually, or use a computer algebra system for fully symbolic differentiation.
The first derivative f'(x) tells you where the function increases or decreases. The second derivative f''(x) tells you how the rate of change itself is changing — this is the curvature or concavity. Together, they provide a complete local picture of the function's behavior: direction (f'), acceleration (f''), and optimal points (where f' = 0 and f'' determines the type).
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