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Derivative & Tangent Line Calculator

Compute instantaneous rates of change, slopes, second derivatives, and tangent lines.

Quick Answer:Select your function type, specify coefficients, and enter x. The solver computes f'(x), f''(x), and generates the tangent equation y = mx + b.

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How to Use the Derivative Calculator: First & Second Derivatives

1Select the function type (polynomial, trigonometric sin/cos, or exponential).
2Configure the multiplier coefficient and power exponent.
3Enter the evaluation point x.
4Read the instantaneous slope f'(x), concavity f''(x), and full tangent equation.

Formula & How It Works

Central Difference Numerical Differentiation

f'(x) ≈ (f(x+h) - f(x-h)) / 2h; f''(x) ≈ (f(x+h) - 2f(x) + f(x-h)) / h²

Calculates symmetric difference quotients with second-order accuracy O(h²).

Worked Examples

Derivative of f(x) = x² at x = 3

Input: f(x) = x², x = 3

Explanation: Analytically d/dx(x²) = 2x; at x=3, 2(3) = 6; tangent passes through (3, 9).

Result: f'(3) = 6.000, Tangent: y = 6x - 9

Frequently Asked Questions

What does the first derivative represent geometrically?▼

The first derivative represents the exact slope of the tangent line touching the curve at that specific coordinate.

What does the second derivative tell you?▼

The second derivative indicates the concavity of the function: positive values indicate concave up (minimum), while negative values indicate concave down.

How is the tangent line equation derived?▼

Using point-slope form y - y₁ = m(x - x₁), where m = f'(x₁) and y₁ = f(x₁).

What step size h is used for central differences?▼

We utilize h = 10⁻⁵, balancing truncation error and floating-point cancellation error.

Quick Reference & Conversion Table

Function f(x)	First Derivative f'(x)	Second Derivative f''(x)	Behavior
x²	2x	2	Parabolic concavity
x³	3x²	6x	Inflection at origin
sin(x)	cos(x)	-sin(x)	Periodic oscillation
e^x	e^x	e^x	Self-replicating slope

⚠️ Common Pitfalls to Avoid

•Evaluating derivatives at points where the function has sharp corners or vertical asymptotes.
•Confusing the derivative value (slope) with the original function's y-value.