Standard derivatives
\[(x^n)'=nx^{n-1},\ (\sin x)'=\cos x,\ (\cos x)'=-\sin x,\ (\tan x)'=\sec^2x\]\[(\sec x)'=\sec x\tan x,\ (\csc x)'=-\csc x\cot x,\ (\cot x)'=-\csc^2x\]
Continuity and Differentiability
Differentiability and the Chain Rule
Understand
The derivative of \(f\) at \(c\) is \(f\,'(c)=\displaystyle\lim_{h\to0}\frac{f(c+h)-f(c)}{h}\), when this limit exists. The left-hand derivative (LHD) and right-hand derivative (RHD) must agree.
Differentiability implies continuity, but not conversely: \(f(x)=|x|\) is continuous at 0, yet LHD \(=-1\) and RHD \(=+1\) – the graph has a corner.
The chain rule
If \(y=f(u)\) and \(u=g(x)\), then \(\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}\). Work from the outside in, multiplying by the derivative of each inner layer.
Example: \(\dfrac{d}{dx}\sin(x^2+1)=\cos(x^2+1)\cdot2x\). Example: \(\dfrac{d}{dx}(3x-2)^5=5(3x-2)^4\cdot3\).
Key Concepts
- Differentiable ⇒ continuous (not conversely).
- Corners (|x − c|) and vertical tangents break differentiability.
- Chain rule: differentiate outer, keep inner, multiply by inner′.
Formula Bank
Chain rule
\[\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\]
Key Points
Continuous but not differentiable
\(|x-c|\) at \(c\); \(x^{1/3}\) at 0 (vertical tangent). Differentiability ⇒ continuity, never the reverse.
Common Mistakes
Forgetting the inner derivative
\(\frac{d}{dx}\sin(3x)=3\cos(3x)\), not \(\cos(3x)\).
Practice & Topic Test
Topic Practice
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Start PracticeTopic Summary
Check LHD = RHD for differentiability at a point; use the chain rule layer by layer for composite functions.
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