Derivative Calculator
Calculate the numerical derivative of a function at a specific point.
Quick Examples
Derivative Rules
The derivative measures the rate of change of a function. It represents the slope of the tangent line at any point.
Basic Differentiation Rules
Constant: d/dx(c) = 0
Sum: d/dx(f+g) = f' + g'
Derivatives of Common Functions
| Function | Derivative |
|---|---|
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec²(x) |
| e^x | e^x |
| ln(x) | 1/x |
| a^x | a^x · ln(a) |
Advanced Rules
- Product Rule: (fg)' = f'g + fg'
- Quotient Rule: (f/g)' = (f'g - fg')/g²
- Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
The Definition: Derivative as a Limit
Formally, the derivative is the limit of the difference quotient:
Geometrically, this is the slope of the tangent line touching the graph at x. For f(x) = x³, expanding (x+h)³ and simplifying gives f'(x) = 3x² — the same answer the power rule produces instantly.
Worked Example: Chain Rule
Differentiate sin(x³) with respect to x:
- Identify outer and inner: outer = sin(u), inner = u = x³.
- Differentiate the outer, leaving the inner untouched: cos(x³).
- Differentiate the inner (power rule): 3x².
- Multiply: 3x² · cos(x³).
Worked Example: Product Rule
Differentiate x² · eˣ:
- Identify u and v: u = x², v = eˣ.
- Find u' and v': u' = 2x, v' = eˣ (eˣ is its own derivative).
- Apply u'v + uv': 2x·eˣ + x²·eˣ.
- Simplify: eˣ(2x + x²).
Worked Example: Quotient Rule
Differentiate (x² + 1)/(x − 1) with respect to x:
- Identify u and v: u = x² + 1, v = x − 1.
- Find u' and v': u' = 2x, v' = 1.
- Apply (u'v − uv')/v²: [2x(x−1) − (x²+1)] / (x−1)².
- Simplify: (x² − 2x − 1) / (x − 1)².
Higher-Order Derivatives
Differentiating twice gives the second derivative f''(x), which describes concavity: positive means the graph curves upward (cup-shaped), negative means it curves downward. Points where concavity changes are inflection points. In physics, if position is f(x), then f'(x) is velocity and f''(x) is acceleration. For f(x) = x³: f'(x) = 3x², f''(x) = 6x — concave down for x < 0, concave up for x > 0, with an inflection point at x = 0.
Implicit Differentiation
When x and y are mixed in an equation like x² + y² = 25, you cannot isolate y easily. Differentiate both sides with respect to x, attaching dy/dx to every y term via the chain rule, then solve: 2x + 2y·dy/dx = 0, so dy/dx = −x/y.
What Derivatives Are Used For
- Optimization: setting the first derivative to zero locates maxima and minima — maximizing profit or minimizing cost.
- Motion: velocity is the derivative of position; acceleration is the derivative of velocity.
- Economics: marginal cost and marginal revenue are derivatives of cost and revenue functions.
- Related rates: how fast a balloon's volume changes as its radius grows, for example.
Derivatives of More Functions
| Function | Derivative |
|---|---|
| 1/x | −1/x² |
| √x | 1/(2√x) |
| sec(x) | sec(x)·tan(x) |
| csc(x) | −csc(x)·cot(x) |
| cot(x) | −csc²(x) |
| arcsin(x) | 1/√(1−x²) |
| arctan(x) | 1/(1+x²) |
Common Mistakes
- Forgetting the inner derivative in the chain rule: d/dx[sin(x³)] is 3x²·cos(x³), not cos(x³).
- Using the power rule on eˣ: its derivative is eˣ, not x·e^(x−1).
- Splitting products term by term: (fg)' is f'g + fg', not f'·g'.
- Quotient rule order: it is (f'g − fg')/g² — the minus sign and the order matter.
- Leaving constants undifferentiated: d/dx[5] = 0, always.
Worked Example: Second Derivative Test
Find and classify the critical points of f(x) = x³ − 3x:
- First derivative: f'(x) = 3x² − 3 = 0 → x = 1 or x = −1.
- Second derivative: f''(x) = 6x.
- At x = 1: f''(1) = 6 > 0 → concave up → local minimum, f(1) = −2.
- At x = −1: f''(−1) = −6 < 0 → concave down → local maximum, f(−1) = 2.
Exponential and Logarithmic Bases
| Function | Derivative |
|---|---|
| aˣ (constant a > 0) | aˣ·ln(a) |
| logₐ(x) | 1 / (x·ln(a)) |
| xˣ | xˣ·(ln(x) + 1) |
The last row comes from logarithmic differentiation: rewrite y = xˣ as ln(y) = x·ln(x), then differentiate implicitly.
Checking Antiderivatives by Differentiating
Differentiation is the fastest way to verify an integral. If someone claims ∫2x·cos(x²) dx = sin(x²) + C, differentiate the answer: d/dx[sin(x²)] = 2x·cos(x²) by the chain rule — confirmed. Whenever an antiderivative looks suspicious, differentiate it; the derivative must return the original integrand.
Notation Guide
| Notation | Name | Reads as |
|---|---|---|
| f'(x) | Lagrange | "f prime of x" |
| dy/dx | Leibniz | "dee y by dee x" |
| d/dx[f(x)] | Leibniz operator | "derivative of f of x" |
| Dₓf | Euler | "D sub x of f" |
Leibniz notation shines for the chain rule — dy/dx = dy/du · du/dx looks like fraction cancellation (though dy/dx is one symbol, not a fraction). Lagrange's f'(x) is compact for higher orders: f'', f'''.
Worked Example: Related Rates
A spherical balloon's radius grows at dr/dt = 2 cm/s. How fast is its volume growing when r = 5 cm?
- Volume: V = (4/3)πr³.
- Differentiate with respect to time (chain rule): dV/dt = 4πr² · dr/dt.
- Substitute: dV/dt = 4π × 25 × 2 = 200π ≈ 628.3 cm³/s.
A Glimpse of Partial Derivatives
For functions of two variables, like f(x, y) = x²y + 3y, the partial derivative ∂f/∂x treats y as a constant: ∂f/∂x = 2xy. Partial derivatives are the foundation of multivariable calculus, optimization in machine learning, and thermodynamics.
Fractional and Negative Exponents
The power rule needs no integer: d/dx[x^(3/2)] = (3/2)x^(1/2), and d/dx[∛x] = d/dx[x^(1/3)] = (1/3)x^(−2/3). Rewrite roots and reciprocals as exponents first, then apply the rule mechanically.
Worked Example: Sum and Difference Rule
Differentiate 4x³ − 2x + 7 term by term:
- 4x³ → 4 × 3x² = 12x² (constant multiple rule).
- −2x → −2.
- 7 → 0 (derivative of a constant).
- Result: 12x² − 2.
From First Principles: One More Look
For f(x) = 1/x, the difference quotient gives [(1/(x+h) − 1/x)]/h = [−h/(x(x+h))]/h = −1/[x(x+h)]. Letting h → 0 yields −1/x² — confirming the power-rule shortcut d/dx[x⁻¹] = −x⁻².
Worked Example: Equation of a Tangent Line
Find the tangent line to f(x) = x² at x = 3:
- Point: f(3) = 9, so the line passes through (3, 9).
- Slope: f'(x) = 2x, so f'(3) = 6.
- Line: y − 9 = 6(x − 3), or y = 6x − 9.
The derivative's first and most direct geometric job is supplying that slope.
Derivatives Worth Memorizing
Beyond the rules, these appear constantly: d/dx[ln(x²+1)] = 2x/(x²+1) by the chain rule, d/dx[x·ln(x)] = ln(x) + 1 by the product rule, and d/dx[e^(−x²)] = −2x·e^(−x²). Recognizing these patterns on sight saves the most time in exams.