Revise Further differentiation: implicit, related rates and L'Hopital for IB Mathematics: Analysis and Approaches (5.7)

Implicit differentiation is what you reach for when a curve is defined by an equation in $x$ and $y$ that you cannot — or would rather not — solve for $y$. The idea is simple: treat $y$ as an unknown function of $x$ and differentiate every term of the equation with respect to $x$. Ordinary terms in $x$ differentiate as usual; every term containing $y$ needs the chain rule, which attaches a factor of $\frac{dy}{dx}$. Mixed terms such as $xy$ need the product rule as well. Once every term is differentiated you are left with a linear equation in $\frac{dy}{dx}$, which you rearrange and solve.

Alongside implicit differentiation, HL expects fluency with a wider family of derivatives. The inverse trigonometric functions have derivatives that are purely algebraic — no trig appears in the answer at all — and the logarithm and exponential composites are constant companions in L'Hopital and related-rates work. Each is really just the chain rule applied to a standard form, so once you know the base derivative you differentiate the inside and multiply.

Inverse trigonometric derivatives: $\dfrac{d}{dx}(\arcsin x) = \dfrac{1}{\sqrt{1-x^2}}, \qquad \dfrac{d}{dx}(\arccos x) = -\dfrac{1}{\sqrt{1-x^2}}, \qquad \dfrac{d}{dx}(\arctan x) = \dfrac{1}{1+x^2}.$ Logarithm and exponential composites (chain rule): $\dfrac{d}{dx}\big(\ln f(x)\big) = \dfrac{f'(x)}{f(x)}, \qquad \dfrac{d}{dx}\big(e^{f(x)}\big) = f'(x)\,e^{f(x)}, \qquad \dfrac{d}{dx}\big(a^{x}\big) = a^{x}\ln a.$

Related rates problems ask how fast one quantity changes given how fast another does. The connection between them is the chain rule: if $y$ depends on $x$ and $x$ depends on time, then $\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}$. In practice you find an equation relating the quantities, differentiate it with respect to $t$, substitute the values at the instant in question, and solve for the rate you want. The recurring trap is a formula with an extra variable — for a cone or a rising liquid level you must first use a geometric constraint (usually similar triangles) to reduce it to the two quantities you care about.

L'Hopital's rule evaluates limits of fractions that first give an indeterminate form — most commonly $\frac{0}{0}$ or $\frac{\infty}{\infty}$. When that happens you may differentiate the numerator and the denominator separately (not as a quotient) and take the limit of the new fraction. The condition is non-negotiable: you must confirm the form is $\frac{0}{0}$ or $\frac{\infty}{\infty}$ first, and if the new fraction is still indeterminate you simply apply the rule again.

If $\displaystyle\lim_{x\to c} f(x)=\lim_{x\to c} g(x)=0$ (or both $\pm\infty$) and $\displaystyle\lim_{x\to c}\dfrac{f'(x)}{g'(x)}$ exists, then $\lim_{x\to c}\dfrac{f(x)}{g(x)}=\lim_{x\to c}\dfrac{f'(x)}{g'(x)}.$

On Paper 1 the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards a correct approach even if the arithmetic later slips; an accuracy mark (A) rewards a correct result and is usually dependent on the method mark being earned. Follow-through (FT) means a correct final step performed on your own earlier (wrong) value still scores, and equivalent correct forms — such as an exact value or its decimal — are accepted. All of that protection exists only if your method is on the page. Study how each mark below is earned by a specific line.

These three techniques share one engine: the chain rule. Implicit differentiation is the chain rule applied to $y$-terms; related rates is the chain rule applied through time; and L'Hopital's rule uses derivatives to compare rates of vanishing. The inverse-trig and log/exponential derivatives you met here reappear constantly — in integration by recognition and substitution, in Maclaurin series, and in differential equations later in the HL calculus option. Master the habit of naming the technique, differentiating with the right rule, and only then substituting, and the harder calculus questions become variations on a method you already own.