Methods of Differentiation — chapter notes

When $x$ and $y$ are mixed, differentiate every term with respect to $x$, treating $y$ as a function of $x$:

• $\dfrac{d}{dx}(y^n) = ny^{n - 1}\dfrac{dy}{dx}$, $\dfrac{d}{dx}(xy) = y + x\dfrac{dy}{dx}$, $\dfrac{d}{dx}e^{y} = e^{y}\dfrac{dy}{dx}$.
• Collect the $\dfrac{dy}{dx}$ terms and divide. For $F(x, y) = 0$ the result is $-\dfrac{F_x}{F_y}$, where $F_x$ treats $y$ as a constant.
• Writing $\dfrac{dy}{dx} = \dfrac{N}{D}$: a horizontal tangent needs $N = 0$, a vertical tangent $D = 0$ (and the point must lie on the curve).

If $x = f(t)$ and $y = g(t)$, then $\dfrac{dy}{dx} = \dfrac{g'(t)}{f'(t)}$ wherever $f'(t) \ne 0$. The answer may stay in terms of $t$.

Useful pairs: $x = a\cos t$, $y = b\sin t$ gives $-\dfrac{b}{a}\cot t$; $x = at^2$, $y = 2at$ gives $\dfrac{1}{t}$. A parameter can describe curves that cross themselves, which no single formula $y = f(x)$ can.

To differentiate $u(x)$ with respect to $v(x)$, divide: $\dfrac{du}{dv} = \dfrac{du/dx}{dv/dx}$, where $\dfrac{dv}{dx} \ne 0$.

Often a substitution shows the answer directly: with $x = \sin\theta$, $\cos^{-1}(1 - 2x^2) = 2\theta = 2\sin^{-1} x$ for $0 < x < 1$, so its derivative with respect to $\sin^{-1} x$ is $2$. Always check the range of the angle before cancelling an inverse function.

For $y = f(x)^{g(x)}$ (with $f(x) > 0$): $\log y = g(x)\log f(x)$, so $\dfrac{dy}{dx} = y\left[g'(x)\log f(x) + g(x)\dfrac{f'(x)}{f(x)}\right]$.

• $\dfrac{d}{dx}x^{x} = x^{x}(1 + \log x)$.
• Sums: split $y = u + v$ and differentiate each term by its own method.
• Long products and quotients: take logs to turn them into sums, e.g. $\dfrac{1}{y}\dfrac{dy}{dx} = \sum \dfrac{n_i}{x - a_i}$ for $y = \prod (x - a_i)^{n_i}$.
• Towers: $x^{x^{x}}$ needs logs twice; the endless tower satisfies $y = x^{y}$, giving $\dfrac{dy}{dx} = \dfrac{y^2}{x(1 - y\log x)}$.

$\dfrac{d^2y}{dx^2} = \dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)$, also written $y''$ or $y_2$. Its sign tells how the slope changes: positive, the curve bends upwards; negative, it bends downwards. (Derivatives beyond the second are not in the ISC 2027 syllabus.)

• Parametric: $\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\left(\dfrac{dy}{dx}\right) \div \dfrac{dx}{dt}$.
• Implicit: differentiate the equation for $y'$ again and substitute $y'$; e.g. for $x^2 + y^2 = r^2$, $y'' = -\dfrac{r^2}{y^3}$.
• Relations: to prove a relation such as $y'' - 4y' + 5y = 0$, find $y'$ and $y''$, then substitute; or clear denominators first and differentiate the product.

Read the form first: mixed $x$ and $y$ means implicit; both given by a parameter means parametric; "with respect to" another function means divide two derivatives; a variable base with a variable exponent, or a long product, means logarithms. Several methods often combine in one question, for example the endless tower is both implicit and logarithmic.