Key Idea: Differentiation rules let you find the gradient of almost any model. Learn the standard derivatives, pick the right rule for products, fractions and nested functions, and link rates with the chain rule.
| Function | Derivative |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| Constant | 0 |
| sin x | cos x |
| cos x | −sin x |
| eˣ | eˣ |
| ln x | 1/x |
| Tangent at x = a | y − f(a) = f′(a)(x − a) |
Which rule when?
- Inside a function? Chain: outer′ × inner′.
- Two factors ×? Product: u′v + uv′.
- A fraction ÷? Quotient: (u′v − uv′)/v².
- Shortcuts: eᵏˣ → k·eᵏˣ and sin(kx) → k·cos(kx).
Related rates
- Link — Write the formula joining the quantities.
- Differentiate — For example dA/dr = 2πr.
- Chain rule — dA/dt = (dA/dr)(dr/dt).
- Substitute last — Put in the value at that instant, then solve.
Differentiate first, and put the numbers in last.
Find the tangent to f(x) = x² − 4x + 5 at x = 3. f′(3) = 2 and f(3) = 2, so y = 2x − 4.
Differentiate h(t) = t·e⁻²ᵗ. h′(t) = e⁻²ᵗ(1 − 2t).
An oil slick's area grows at 8 m² s⁻¹. How fast is r growing when r = 20 m? 8 = 40π dr/dt, so dr/dt = 1/(5π) ≈ 0.0637 m s⁻¹.
A balloon is shrinking. What sign does dV/dt have? Negative.
Which angle unit must x be in for the sin and cos derivatives? Radians.