Implicit Differentiation — Last Minute Summary

$\frac{dy}{dx}$ in 4 Steps

1. Differentiate every term w.r.t. $x$
2. For $y$-terms: multiply by $\frac{dy}{dx}$
3. For $xy$-terms: product rule $y + x\frac{dy}{dx}$
4. Rearrange: collect $\frac{dy}{dx}$ terms, factor, divide

$\frac{d^2y}{dx^2}$ in 3 Steps

1. Differentiate $\frac{dy}{dx}$ w.r.t. $x$ (use quotient rule if fraction)
2. Replace any $\frac{dy}{dx}$ with the expression from step 1
3. Simplify using the original equation to get answer in $x$, $y$ only

Tangent and Normal

• Tangent: $y - y_0 = m(x - x_0)$ where $m = \frac{dy}{dx}\big|_{(x_0,y_0)}$
• Normal: $y - y_0 = -\frac{1}{m}(x - x_0)$ (if $m \neq 0$)

Common Derivatives

$\frac{d}{dx}(y^n) = n y^{n-1}\frac{dy}{dx}$

$\frac{d}{dx}(\sin y) = \cos y\frac{dy}{dx}$

$\frac{d}{dx}(e^y) = e^y\frac{dy}{dx}$

$\frac{d}{dx}(\ln y) = \frac{1}{y}\frac{dy}{dx}$

$\frac{d}{dx}(xy) = y + x\frac{dy}{dx}$

Stationary Points

Set $\frac{dy}{dx} = 0$ → solve simultaneously with original equation.

Trap Checklist

❌ Forgetting $\frac{dy}{dx}$ on $y$-terms

❌ Product rule on $xy$: $\frac{d}{dx}(xy) = y + x\frac{dy}{dx}$, not just $y$

❌ $\frac{d^2y}{dx^2}$ substitution: substitute $\frac{dy}{dx}$ before simplifying

❌ Using $x$ and $y$ from the wrong point