Derivatives, Differentials

Definition of a Derivative

If $y = f(x)$, the derivative of $y$ or $f(x)$ with respect to $x$ is defined as

$\displaystyle \frac{dy}{dx} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$

where $h = \Delta x$. The derivative is also denoted by $y'$, $\frac{df}{dx}$ or $f'(x)$. The process of taking a derivative is called differentiation.

General Rules of Differentiation

In the following, $u, v, w$ are functions of $x$; $a, b, c, n$ are constants [restricted if indicated]; $e = 2.71828\ldots$ is the natural base of logarithms; $\ln u$ is the natural logarithm of $u$ [i.e. the logarithm to the base $e$] where it is assumed that $u > 0$ and all angles are in radians.

$\displaystyle \frac{d}{dx}(c) = 0$

$\displaystyle \frac{d}{dx}(cx) = c$

$\displaystyle \frac{d}{dx}(cx^n) = ncx^{n-1}$

$\displaystyle \frac{d}{dx}(u \pm v \pm w \pm \cdots) = \frac{du}{dx} \pm \frac{dv}{dx} \pm \frac{dw}{dx} \pm \cdots$

$\displaystyle \frac{d}{dx}(cu) = c\frac{du}{dx}$

$\displaystyle \frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$

$\displaystyle \frac{d}{dx}(uvw) = uv\frac{dw}{dx} + uw\frac{dv}{dx} + vw\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\,(du/dx) - u\,(dv/dx)}{v^2}$

$\displaystyle \frac{d}{dx}(u^n) = nu^{n-1}\frac{du}{dx}$

$\displaystyle \frac{dy}{dx} = \frac{dy}{du}\,\frac{du}{dx}$ (Chain rule)

$\displaystyle \frac{du}{dx} = \frac{1}{dx/du}$

$\displaystyle \frac{dy}{dx} = \frac{dy/du}{dx/du}$

Derivatives of Trigonometric and Inverse Trigonometric Functions

$\displaystyle \frac{d}{dx}\sin u = \cos u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\cos u = -\sin u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\tan u = \sec^2 u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\cot u = -\csc^2 u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\sec u = \sec u\,\tan u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\csc u = -\csc u\,\cot u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\sin^{-1} u = \frac{1}{\sqrt{1 - u^2}}\,\frac{du}{dx}$ $\left[-\frac{\pi}{2} \lt \sin^{-1} u \lt \frac{\pi}{2}\right]$

$\displaystyle \frac{d}{dx}\cos^{-1} u = \frac{-1}{\sqrt{1 - u^2}}\,\frac{du}{dx}$ $[0 \lt \cos^{-1} u \lt \pi]$

$\displaystyle \frac{d}{dx}\tan^{-1} u = \frac{1}{1 + u^2}\,\frac{du}{dx}$ $\left[-\frac{\pi}{2} \lt \tan^{-1} u \lt \frac{\pi}{2}\right]$

$\displaystyle \frac{d}{dx}\cot^{-1} u = \frac{-1}{1 + u^2}\,\frac{du}{dx}$ $[0 \lt \cot^{-1} u \lt \pi]$

$\displaystyle \frac{d}{dx}\sec^{-1} u = \frac{1}{|u|\sqrt{u^2 - 1}}\,\frac{du}{dx} = \frac{\pm 1}{u\sqrt{u^2 - 1}}\,\frac{du}{dx}$ $\left[\begin{array}{ll} + & \text{if } 0 \lt \sec^{-1} u \lt \pi/2 \\ - & \text{if } \pi/2 \lt \sec^{-1} u \lt \pi \end{array}\right]$

$\displaystyle \frac{d}{dx}\csc^{-1} u = \frac{-1}{|u|\sqrt{u^2 - 1}}\,\frac{du}{dx} = \frac{\mp 1}{u\sqrt{u^2 - 1}}\,\frac{du}{dx}$ $\left[\begin{array}{ll} - & \text{if } 0 \lt \csc^{-1} u \lt \pi/2 \\ + & \text{if } -\pi/2 \lt \csc^{-1} u \lt 0 \end{array}\right]$

Derivatives of Exponential and Logarithmic Functions

$\displaystyle \frac{d}{dx}\log_a u = \frac{\log_a e}{u}\,\frac{du}{dx}$ $a > 0,\ a \ne 1$

$\displaystyle \frac{d}{dx}\ln u = \frac{d}{dx}\log_e u = \frac{1}{u}\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}a^u = a^u \ln a\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}e^u = e^u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}u^v = \frac{d}{dx}e^{v\ln u} = e^{v\ln u}\,\frac{d}{dx}[v\ln u] = vu^{v-1}\frac{du}{dx} + u^v \ln u\,\frac{dv}{dx}$

Derivatives of Hyperbolic and Inverse Hyperbolic Functions

$\displaystyle \frac{d}{dx}\sinh u = \cosh u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\cosh u = \sinh u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\tanh u = \operatorname{sech}^2 u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\coth u = -\operatorname{csch}^2 u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\operatorname{sech} u = -\operatorname{sech} u\,\tanh u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\operatorname{csch} u = -\operatorname{csch} u\,\coth u\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\sinh^{-1} u = \frac{1}{\sqrt{u^2 + 1}}\,\frac{du}{dx}$

$\displaystyle \frac{d}{dx}\cosh^{-1} u = \frac{\pm 1}{\sqrt{u^2 - 1}}\,\frac{du}{dx}$ $\left[\begin{array}{ll} + & \text{if } \cosh^{-1} u > 0 \\ - & \text{if } \cosh^{-1} u \lt 0 \end{array}\right],\ \allowbreak {u > 1}$

$\displaystyle \frac{d}{dx}\tanh^{-1} u = \frac{1}{1 - u^2}\,\frac{du}{dx}$ $[-1 \lt u \lt 1]$

$\displaystyle \frac{d}{dx}\coth^{-1} u = \frac{1}{1 - u^2}\,\frac{du}{dx}$ $[u > 1 \text{ or } u \lt -1]$

$\displaystyle \frac{d}{dx}\operatorname{sech}^{-1} u = \frac{\mp 1}{u\sqrt{1 - u^2}}\,\frac{du}{dx}$ $\left[\begin{array}{ll} - & \text{if } \operatorname{sech}^{-1} u > 0 \\ + & \text{if } \operatorname{sech}^{-1} u \lt 0 \end{array}\right],\ \allowbreak {0 \lt u \lt 1}$

$\displaystyle \frac{d}{dx}\operatorname{csch}^{-1} u = \frac{-1}{|u|\sqrt{1 + u^2}}\,\frac{du}{dx} = \frac{\mp 1}{u\sqrt{1 + u^2}}\,\frac{du}{dx}$ $[-\text{ if } u > 0,\ +\text{ if } u \lt 0]$

Higher Derivatives

The second, third and higher derivatives are defined as follows.

Second derivative $\displaystyle = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d^2y}{dx^2} = f''(x) = y''$

Third derivative $\displaystyle = \frac{d}{dx}\left(\frac{d^2y}{dx^2}\right) = \frac{d^3y}{dx^3} = f'''(x) = y'''$

$n$-th derivative $\displaystyle = \frac{d}{dx}\left(\frac{d^{n-1}y}{dx^{n-1}}\right) = \frac{d^ny}{dx^n} = f^{(n)}(x) = y^{(n)}$

Leibniz's Rule for Higher Derivatives of Products

Let $D^p$ stand for the operator $\dfrac{d^p}{dx^p}$, so that $D^p u = \dfrac{d^pu}{dx^p}$ is the $p$-th derivative of $u$. Then

$\displaystyle D^n(uv) = uD^nv + \binom{n}{1}(Du)(D^{n-1}v) + \binom{n}{2}(D^2u)(D^{n-2}v) + \cdots + vD^nu$

where $\dbinom{n}{1}, \dbinom{n}{2}, \ldots$ are the binomial coefficients.

As special cases we have

$\displaystyle \frac{d^2}{dx^2}(uv) = u\frac{d^2v}{dx^2} + 2\frac{du}{dx}\frac{dv}{dx} + v\frac{d^2u}{dx^2}$

$\displaystyle \frac{d^3}{dx^3}(uv) = u\frac{d^3v}{dx^3} + 3\frac{du}{dx}\frac{d^2v}{dx^2} + 3\frac{d^2u}{dx^2}\frac{dv}{dx} + v\frac{d^3u}{dx^3}$

Differentials

Let $y = f(x)$ and $\Delta y = f(x + \Delta x) - f(x)$. Then

$\displaystyle \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x} = f'(x) + \varepsilon = \frac{dy}{dx} + \varepsilon$

where $\varepsilon \to 0$ as $\Delta x \to 0$. Thus

$\Delta y = f'(x)\,\Delta x + \varepsilon\,\Delta x$

If we call $\Delta x = dx$ the differential of $x$, then we define the differential of $y$ to be

$dy = f'(x)\,dx$

Rules for Differentials

The rules for differentials are exactly analogous to those for derivatives. As examples we observe that

$d(u \pm v \pm w \pm \cdots) = du \pm dv \pm dw \pm \cdots$

$d(uv) = u\,dv + v\,du$

$\displaystyle d\left(\frac{u}{v}\right) = \frac{v\,du - u\,dv}{v^2}$

$d(u^n) = nu^{n-1}\,du$

$d(\sin u) = \cos u\,du$

$d(\cos u) = -\sin u\,du$

Partial Derivatives

Let $f(x, y)$ be a function of the two variables $x$ and $y$. Then we define the partial derivative of $f(x, y)$ with respect to $x$, keeping $y$ constant, to be

$\displaystyle \frac{\partial f}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x}$

Similarly the partial derivative of $f(x, y)$ with respect to $y$, keeping $x$ constant, is defined to be

$\displaystyle \frac{\partial f}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y) - f(x, y)}{\Delta y}$

Partial derivatives of higher order can be defined as follows.

$\displaystyle \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right),$ $\displaystyle \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right)$

$\displaystyle \frac{\partial^2 f}{\partial x\,\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right),$ $\displaystyle \frac{\partial^2 f}{\partial y\,\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)$

The mixed partial derivatives $\frac{\partial^2 f}{\partial x\,\partial y}$ and $\frac{\partial^2 f}{\partial y\,\partial x}$ will be equal if the function and its partial derivatives are continuous, i.e. in such a case the order of differentiation makes no difference.

The differential of $f(x, y)$ is defined as

$\displaystyle df = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy$

where $dx = \Delta x$ and $dy = \Delta y$.

Extensions to functions of more than two variables are exactly analogous.