Derivative Calculator
Differentiate algebraic, trigonometric, exponential, and logarithmic functions with step-by-step rule breakdowns, higher-order derivatives (f', f'', f'''), and live tangent line plotting at any point x₀.
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Step-by-Step Differentiation Derivation
How to Differentiate a Function f(x)
To find the derivative f'(x), apply standard calculus rules: 1. Power Rule: d/dx[xⁿ] = n·xⁿ⁻¹. 2. Product Rule: (uv)' = u'v + uv'. 3. Quotient Rule: (u/v)' = (u'v − uv') / v². 4. Chain Rule: [f(g(x))]' = f'(g(x)) · g'(x). The derivative gives the exact slope of the tangent line y − f(x₀) = f'(x₀)(x − x₀) at point x₀.
The Limit Definition of the Derivative
The derivative represents the instantaneous rate of change of a continuous function $f(x)$. Developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century:
Geometrically, as the secant interval h approaches 0, the secant line through (x, f(x)) and (x+h, f(x+h)) converges to the unique tangent line touching the curve at x.
The Fundamental Rules of Differential Calculus
Applies to positive, negative, and fractional exponents (e.g. d/dx[x⁻²] = −2x⁻³).
Used when multiplying two variable functions (e.g. x² · sin(x)).
"Low D-High minus High D-Low, over the square of what's below."
Essential for composite expressions like (3x² + 5)⁴ or sin(2x).
Derivatives of Transcendental Functions
| Function f(x) | Derivative f'(x) | Chain Rule Generalization |
|---|---|---|
| sin(x) | cos(x) | cos(u) · u' |
| cos(x) | −sin(x) | −sin(u) · u' |
| tan(x) | sec²(x) | sec²(u) · u' |
| eˣ | eˣ | eᵘ · u' |
| ln(x) | 1 / x | u' / u |
Higher-Order Derivatives & Concavity Analysis
Differentiating repeatedly produces higher-order derivatives with direct physical and geometric meanings:
f'(x) = 0 identifies critical points. If f' changes from + to −, it is a local maximum; if − to +, a local minimum.
f''(x) > 0 indicates concave upward (∪); f''(x) < 0 indicates concave downward (∩). f''(x) = 0 identifies inflection points.
Real-World Applications of Differential Calculus
Classical Physics & Kinematics
Position $s(t)$, velocity $v(t) = s'(t)$, acceleration $a(t) = s''(t)$, and jerk $j(t) = s'''(t)$ govern all Newtonian mechanics.
Economics & Marginal Analysis
Economists use marginal cost $MC(q) = C'(q)$ and marginal revenue $MR(q) = R'(q)$ to solve profit maximization where $MR = MC$.
AI & Neural Network Training
Modern deep learning backpropagation uses the multivariable chain rule to compute gradients $\nabla L$ across billions of transformer parameters.
Step-by-Step Worked Numerical Solutions
Problem: Find the derivative and tangent line equation for f(x) = x² · eˣ at x₀ = 1.
Common Pitfalls & Mistakes
Forgetting the Inner Chain Rule
The derivative of sin(3x) is 3cos(3x), NOT cos(3x). You must multiply by the inner derivative d/dx[3x] = 3.
Differentiating Products Term-by-Term
d/dx[u · v] is NOT u' · v'. You must use the product rule u'v + uv'.
Wrong Sign in Quotient Rule
The numerator is (u'v − uv'), not (uv' − u'v). Subtraction order matters strictly.
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Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.