Engineering Services (ESE) – Mathematics
Topic: Differentiation
Differentiation is the process of finding the rate of change of a function.
In engineering this tells you how a physical quantity changes with respect to another – e.g., velocity from position, stress from strain, etc.
1. Simple Explanation
| Concept | What it means | Engineering intuition |
|---|---|---|
Derivative f'(x) |
Slope of the tangent line to y = f(x) at a point x. |
How fast a quantity changes at a particular instant. |
| Limit definition | f'(x) = lim_{h→0} [f(x+h) – f(x)] / h |
Think of it as “instantaneous rate” rather than average. |
| Notation | df/dx, f'(x), ∂f/∂x (partial derivative for multivariable). |
Pick one; all mean the same in single‑variable problems. |
2. Key Points (Quick Reference)
| Topic | Formula / Rule | Typical Use |
|---|---|---|
| Power Rule | d/dx [xⁿ] = n·xⁿ⁻¹ |
Any polynomial term |
| Constant Rule | d/dx [c] = 0 |
Removes constants |
| Sum/Difference Rule | d/dx [u ± v] = u' ± v' |
Combine derivatives |
| Product Rule | d/dx [u·v] = u'·v + u·v' |
For products like x·sin x |
| Quotient Rule | d/dx [u/v] = (u'·v – u·v') / v² |
For divisions |
| Chain Rule | d/dx [g(f(x))] = g'(f(x))·f'(x) |
Composite functions (e.g., sin(x²) ) |
| Exponential & Logarithmic | d/dx e^x = e^x; d/dx ln x = 1/x |
Common in growth/decay |
| Trigonometric | d/dx sin x = cos x; d/dx cos x = –sin x; d/dx tan x = sec²x |
Oscillatory phenomena |
| Inverse Trig | d/dx arcsin x = 1/√(1–x²) |
Often appear in kinematics |
| Implicit Differentiation | Differentiate both sides w.r.t x |
When y is defined implicitly (e.g., x² + y² = r²) |
| Higher‑order Derivatives | f''(x) = d/dx[f'(x)] |
Curvature, acceleration |
| Physical Units | Keep track: if y is meters, x seconds → f'(x) m/s |
Avoid unit errors |
3. Worked Example
Problem:
Differentiate the function
[
f(x) = x^2 \sin x
]
Step‑by‑Step
-
Identify the structure.
f(x)is a product: (u(x)=x^2) and (v(x)=\sin x). -
Apply Product Rule
[ f'(x) = u'(x),v(x) + u(x),v'(x) ] -
Compute each part
- (u'(x) = d/dx[x^2] = 2x) (Power Rule)
- (v'(x) = d/dx[\sin x] = \cos x) (Trigonometric Rule)
-
Substitute back
[ f'(x) = (2x)(\sin x) + (x^2)(\cos x) ] -
Simplify
[ f'(x) = 2x\sin x + x^2\cos x ]
Result
[
\boxed{,f'(x) = 2x\sin x + x^2\cos x,}
]
Engineering Interpretation
If x is time (seconds) and f(x) is a displacement (meters), then f'(x) gives the instantaneous velocity at time x. The two terms represent contributions from the changing amplitude (x²) and the oscillatory part (sin x).
4. Quick Revision Summary
- Derivative = Slope of tangent – instantaneous rate of change.
- Key Rules to Memorise
- Power, Constant, Sum/Diff, Product, Quotient, Chain.
- Exponential, Logarithmic, Trigonometric, Inverse Trig.
- Process
- Spot the rule needed.
- Apply it, simplifying step by step.
- Check units: result must have correct physical units.
- Common Pitfalls
- Forgetting the product or quotient term.
- Mixing up chain rule signs.
- Ignoring constants (they vanish).
- Practice – Differentiate functions like
- (f(x)=e^{x^2}) (chain rule)
- (g(x)=\frac{\ln x}{x}) (quotient rule)
- (h(x)=x^3\cos x) (product rule)
- Implicit example: (x^2 + y^2 = 25).
Tip: Always write the derivative in a simplified form; this helps when you need to evaluate it later (e.g., at a specific point or for integration).
Final Checklist for Differentiation Tasks
- Identify if the function is a single variable or implicit.
- Choose the appropriate rule(s).
- Apply the rule(s) carefully,