Differentiation 1

Derivative definition. For $f$ differentiable at $x$:

$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.$

Worked example. Find $f'(x)$ from first principles for $f(x) = x^2$.

$f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h) = 2x.$

Worked example. Find $f'(x)$ from first principles for $f(x) = 1/x$.

$\frac{f(x + h) - f(x)}{h} = \frac{1/(x+h) - 1/x}{h} = \frac{-h/[x(x+h)]}{h} = \frac{-1}{x(x+h)}.$

Taking $h \to 0$: $f'(x) = -1/x^2$. ✓ matches power rule.

• First principles tests rigorous algebra.
• Always show the limit explicitly.

Chain rule. If $y = f(g(x))$, then $y' = f'(g(x)) \cdot g'(x)$.

Product rule. $(uv)' = u'v + uv'$.

Quotient rule. $(u/v)' = (u'v - uv')/v^2$.

Worked example (chain). Differentiate $y = \sin(3x^2 + 1)$.

Inner $u = 3x^2 + 1$, $du/dx = 6x$. $dy/du = \cos u$. So $dy/dx = 6x\cos(3x^2 + 1)$.

Worked example (product). $y = x^2 e^x$.

$y' = 2x \cdot e^x + x^2 \cdot e^x = e^x(2x + x^2) = x e^x(2 + x)$.

Worked example (quotient). $y = (2x + 1)/(x^2 + 3)$.

$y' = \frac{2(x^2 + 3) - (2x + 1)(2x)}{(x^2 + 3)^2} = \frac{2x^2 + 6 - 4x^2 - 2x}{(x^2 + 3)^2} = \frac{-2x^2 - 2x + 6}{(x^2 + 3)^2}$.

Tangent at $(a, f(a))$: $y - f(a) = f'(a)(x - a)$.

Normal: gradient $= -1/f'(a)$.

• Chain: differentiate outer, KEEP inner, times derivative of inner.
• Product/quotient: pick u, v carefully and stay neat.
• Tangent and normal: perpendicular gradients multiply to $-1$.

Outer: $\ln u \Rightarrow 1/u$. Middle: $\cos v \Rightarrow -\sin v$. Inner: $2x \Rightarrow 2$.