Revision sheet

Implicit differentiation

1. Differentiate both sides with respect to $x$.
2. Each $y$ term gets a factor $y'$; use the product rule for $xy$.
3. Solve for $y'$.

Watch out: $(y^2)' = 2y\,y'$, not $2y$.

Tangents to implicit curves

• Slope: substitute the point into $y'$ (both $x$ and $y$).
• Line: $y = y_0 + m(x - x_0)$.

Watch out: the tangent to a circle is perpendicular to the radius.

Derivative of an inverse

1. Find $a = f^{-1}(b)$.
2. Compute $f'(a)$.
3. $(f^{-1})'(b) = 1 / f'(a)$.

Watch out: evaluate $f'$ at $a$, the input of $f$, not at $b$.

Related rates

1. Equation linking the quantities.
2. Differentiate with respect to $t$.
3. Substitute the values at that moment; solve for the unknown rate.

Watch out: substitute changing values only after differentiating.

Right-triangle rates

• $x^2 + y^2 = D^2 \Rightarrow x\,x' + y\,y' = D\,D'$.
• Find the missing side at that moment first.

Watch out: rates at right angles do not add; decreasing lengths have negative rates.

Similar triangles

1. Use the fixed ratio to write one variable in terms of another.
2. Differentiate with respect to $t$.
3. Substitute the moment's values.

Watch out: the big triangle's base runs from the lamp to the shadow's tip.