Derivatives and Applications

What is a Derivative?

The derivative measures the instantaneous rate of change of a function:

f'(x) = lim[h→0] (f(x+h) - f(x)) / h

Geometrically: the slope of the tangent line at a point.

Basic Differentiation Rules

| Rule | Formula | |------|---------| | Constant | d/dx(c) = 0 | | Power | d/dx(xⁿ) = nxⁿ⁻¹ | | Sum | (f+g)' = f' + g' | | Product | (fg)' = f'g + fg' | | Quotient | (f/g)' = (f'g - fg')/g² | | Chain | d/dx[f(g(x))] = f'(g(x))×g'(x) |

Common Derivatives

| f(x) | f'(x) | |------|-------| | sin x | cos x | | cos x | -sin x | | tan x | sec²x | | eˣ | eˣ | | ln x | 1/x | | aˣ | aˣ ln a |

Higher Order Derivatives

• f''(x): second derivative (concavity, acceleration)
• f'''(x): third derivative
• dⁿy/dxⁿ: nth derivative

Applications

Finding Extrema

1. Find critical points: f'(x) = 0 or undefined
2. Second derivative test: f''(x) > 0 → min; f''(x) < 0 → max

Optimization Example

Maximize the area of a rectangle with perimeter = 40 m:

• P = 2l + 2w = 40 → l = 20 - w
• A = lw = (20-w)w = 20w - w²
• dA/dw = 20 - 2w = 0 → w = 10, l = 10 (square!)
• A_max = 100 m²

Related Rates

If A = πr²: dA/dt = 2πr × dr/dt

Useful for: pipe flows, expanding shapes, moving objects.

Engineering Uses

• Velocity v = ds/dt; Acceleration a = dv/dt
• Beam deflection slope = dy/dx
• Shear force V = dM/dx
• Load intensity w = dV/dx

Summary

Derivatives are everywhere in engineering equations. Master the power rule and chain rule first. Then practice optimization and related rates — they're on every FE exam.