Differentiation

Leibniz Formula for Products

For $y = u(x)v(x)$, the $n$th derivative is:

$\frac{d^n y}{dx^n} = \sum_{k=0}^{n} \binom{n}{k} \frac{d^{n-k}u}{dx^{n-k}} \frac{d^k v}{dx^k}$

This generalises the product rule to higher derivatives.

Example: If $y = x^2 e^x$, find $\frac{d^3 y}{dx^3}$.

Let $u = x^2$ and $v = e^x$. Then:

• $u' = 2x$, $u'' = 2$, $u''' = 0$
• $v' = v'' = v''' = e^x$

$y''' = u'''v + 3u''v' + 3u'v'' + uv''' = 0 + 6e^x + 6xe^x + x^2e^x = (x^2 + 6x + 6)e^x$

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Curve Sketching

Key techniques for sketching curves $y = f(x)$:

1. Intercepts: Where does $f(x) = 0$? Where does $y = 0$?
2. Asymptotes:
   • Horizontal: $\lim_{x \to \pm\infty} f(x)$
   • Vertical: Where $f(x) \to \infty$
   • Oblique: Divide polynomials for rational functions
3. Stationary points: Where $f'(x) = 0$
   • Local maximum: $f'' < 0$
   • Local minimum: $f'' > 0$
   • Point of inflection: $f'' = 0$ but derivative doesn’t change sign
4. Symmetry: Even ($f(-x) = f(x)$) or odd ($f(-x) = -f(x)$)?
5. Behaviour at special points: Near asymptotes, near the origin

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Newton-Raphson Method

To find roots of $f(x) = 0$, iterate:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Geometric interpretation: Follow the tangent line to the $x$-axis.

Convergence: Quadratic (very fast) near a root. But can fail if:

• $f'(x_n) = 0$ (horizontal tangent)
• Starting point is poorly chosen (cycle or divergence)

Example: Estimate $\sqrt{2}$ by solving $f(x) = x^2 - 2 = 0$.

$x_{n+1} = x_n - \frac{x_n^2 - 2}{2x_n} = x_n - \frac{x_n}{2} + \frac{1}{x_n} = \frac{x_n}{2} + \frac{1}{x_n}$

Starting with $x_0 = 1$: $x_1 = 1.5$, $x_2 \approx 1.417$, $x_3 \approx 1.414$.

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Implicit Differentiation

For functions defined implicitly by $F(x, y) = 0$:

$\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y}$

Or differentiate both sides with respect to $x$, treating $y$ as a function of $x$.

Example: Find $\frac{dy}{dx}$ for $x^2 + y^2 = a^2$.

Differentiating: $2x + 2y\frac{dy}{dx} = 0$, so $\frac{dy}{dx} = -\frac{x}{y}$.

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Parametric Differentiation

If $x = x(t)$ and $y = y(t)$:

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

For the second derivative: $\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx} = \frac{d}{dt}\left(\frac{y'}{x'}\right) \cdot \frac{1}{x'}$

Example: For $x = \cos t$, $y = \sin t$:

$\frac{dy}{dx} = \frac{\cos t}{-\sin t} = -\cot t$

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Logarithmic Differentiation

For products/quotients/powers, take logarithms first:

$\text{If } y = \frac{u^a \cdot v^b}{w^c}, \text{ then } \ln y = a\ln u + b\ln v - c\ln w$

Differentiate: $\frac{y'}{y} = \frac{au'}{u} + \frac{bv'}{v} - \frac{cw'}{w}$

Example: Find $\frac{d}{dx}(x^x)$.

$y = x^x$, so $\ln y = x\ln x$. Then: $\frac{y'}{y} = \ln x + 1 \Rightarrow y' = x^x(\ln x + 1)$

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Higher Derivatives and Patterns

Some functions have recognisable derivative patterns:

$f(x)$	$f'(x)$	$f''(x)$	$f^{(n)}(x)$
$e^x$	$e^x$	$e^x$	$e^x$
$\sin x$	$\cos x$	$-\sin x$	$\sin(x + n\pi/2)$
$\cos x$	$-\sin x$	$-\cos x$	$\cos(x + n\pi/2)$
$\ln x$	$1/x$	$-1/x^2$	$(-1)^{n-1}(n-1)!/x^n$