MISSION · Calculus
The Rules for Differentiating
The shortcuts that replace the limit, and when each applies
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The idea in one sentence
Every derivative could be found from the limit definition, and almost none are — because a handful of rules cover everything you’ll meet.
The power rule
\[\frac{d}{dx}x^n = n x^{n-1}\]Bring the power down in front, then reduce it by one. \(x^5\) becomes \(5x^4\); \(x^2\) becomes \(2x\); \(x\) becomes 1.
It works for every exponent, not just whole ones. \(\sqrt{x} = x^{1/2}\) differentiates to \(\frac{1}{2}x^{-1/2}\), and \(\frac{1}{x} = x^{-1}\) becomes \(-x^{-2}\). Rewriting roots and fractions as powers first is the standard opening move.
A constant differentiates to zero, which the rule delivers if you write \(7\) as \(7x^0\). Geometrically it’s obvious: a constant function is a flat line, and flat means slope zero.
Sums, constants, products, quotients
Sums come apart term by term: differentiate each piece and add. So \(3x^4 - 2x + 7\) becomes \(12x^3 - 2\).
Constant multiples come along for the ride: \(\frac{d}{dx}[5f(x)] = 5f'(x)\).
Products do not. \(\frac{d}{dx}[f \cdot g]\) is not \(f' \cdot g'\) — a tempting symmetry that’s simply false. The correct rule:
\[(fg)' = f'g + fg'\]Quotients follow from it:
\[\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}\]The order matters in the numerator — subtraction isn’t symmetric, and swapping the terms flips the sign of every answer.
The chain rule
For a function inside a function — \((3x + 1)^5\), \(\sin(x^2)\), \(e^{4x}\) — differentiate the outside, keeping the inside untouched, then multiply by the derivative of the inside.
\[\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)\]So \((3x+1)^5\) gives \(5(3x+1)^4 \cdot 3 = 15(3x+1)^4\). That final \(\times 3\) is the step everyone forgets, and it’s the whole content of the rule.
The name describes the mechanism: rates multiply along a chain. If \(y\) changes 5 times as fast as \(u\), and \(u\) changes 3 times as fast as \(x\), then \(y\) changes 15 times as fast as \(x\).
Choosing the right rule
Look at the structure before differentiating anything:
- Terms added or subtracted → handle each separately.
- Two things multiplied → product rule.
- One thing over another → quotient rule.
- Something wrapped inside something else → chain rule.
Complicated expressions need several, applied from the outside in. The most common failure isn’t arithmetic, it’s misreading which structure you’re looking at.
Common mix-ups
- Differentiating a product as \(f'g'\). It isn’t.
- Dropping the inside derivative in the chain rule.
- Flipping the quotient rule’s numerator, which negates the answer.
- Forgetting to rewrite roots as powers before applying the power rule.
Check yourself
Differentiate \(y = (2x^2 + 1)^3\) and \(y = x^2 \sin x\).
Chain rule: \(3(2x^2+1)^2 \cdot 4x = 12x(2x^2+1)^2\). Product rule: \(2x\sin x + x^2\cos x\).
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