Differentiation

Differentiation:

The derivative of a function is a measure of how the function changes as a result of a change in the value of its argument. Given the function $f (x)$ , the derivative of f with respect to $x$ is written as $\frac{d f (x)}{d x}$ or as $\frac{d f}{d x} (x)$ , and is defined by

\frac{d f}{d x} (x) = \lim_{Δ x \to 0} \frac{f (x + Δ x) - f (x)}{Δ x}

As shown in the figure, the derivative of the function $f (x)$ at point $x$ gives the slope of the function at $x$ in terms of the ratio of the rise divided by the run for the line $A B$ that is tangent to the curve at point $x$ .

Graph of y = f(x). The secant connects the curve at x and x plus delta x, with horizontal run delta x and vertical rise delta f. The dashed tangent AB touches the curve at x. Its slope triangle has a run of 1 and a rise equal to df/dx at x.

The derivative of the function $f (x)$ is also sometimes written as $f' (x)$ . As shown in the figure, one can also write the definition of the derivative as

f' (x) = \lim_{Δ x \to 0} \frac{Δ f}{Δ x}

The basic rules of differentiation are

\begin{array}{l} \frac{d}{d x} [a f (x)] = a \frac{d}{d x} [f (x)] where a is a constant \\ \frac{d}{d x} [f (x) + g (x)] = \frac{d}{d x} [f (x)] + \frac{d}{d x} [g (x)] \\ \frac{d}{d x} [f (x) g (x)] = \frac{d}{d x} [f (x)] g (x) + f (x) \frac{d}{d x} [g (x)] \end{array}

The derivative of commonly used functions:

The following is a list of the derivatives of some of the more commonly used functions.

\begin{array}{l} \frac{d}{d x} (constant) = 0 \\ \frac{d}{d x} (x) = 1 \\ \frac{d}{d x} (x^{2}) = 2 x \\ \frac{d}{d x} (\frac{1}{x}) = - \frac{1}{x^{2}} \\ \frac{d}{d x} (x^{n}) = n x^{n - 1} \end{array}

\begin{array}{l} \frac{d}{d x} (\sin x) = \cos x \\ \frac{d}{d x} (\cos x) = - \sin x \\ \frac{d}{d x} (\tan x) = \sec^{2} x \end{array}

\begin{array}{l} \frac{d}{d x} (\ln (x)) = \frac{1}{x} \\ \frac{d}{d x} (e^{x}) = e^{x} \end{array}

The product rule for derivatives:

Consider a function such as $f (x) = g (x) h (x)$ that is the product of two functions. The product rule can be used to calculate the derivative of f with respect to x. The product rule states that

\frac{d}{d x} (g h) = \frac{d g}{d x} h + g \frac{d h}{d x}

For example, to take the derivative of $f (x) = (2 x + 3) {(4 x + 5)}^{2}$ one can follow these steps

\begin{array}{l} \frac{d f}{d x} = \frac{d}{d x} [(2 x + 3) {(4 x + 5)}^{2}] \\ = \frac{d (2 x + 3)}{d x} {(4 x + 5)}^{2} + (2 x + 3) \frac{d {(4 x + 5)}^{2}}{d x} \\ = 2 {(4 x + 5)}^{2} + (2 x + 3) [2 (4) (4 x + 5)] \\ = 2 {(4 x + 5)}^{2} + 8 (2 x + 3) (4 x + 5) \\ = 2 (12 x + 17) (4 x + 5) \end{array}

The chain rule for derivatives:

Consider the function $f (U)$ , where U is a function of x. One can calculate the derivative of f with respect to x by using the chain rule given by

\frac{d f [U (x)]}{d x} = \frac{d f (U)}{d U} \frac{d U}{d x}

For example, to calculate the derivative of $f (U) = U^{n}$ with respect to x, where $U = x^{2} + a$ , one can follow these steps

\begin{array}{l} \frac{d f}{d U} = \frac{d U^{n}}{d U} = n U^{n - 1} \\ \frac{d U}{d x} = \frac{d (x^{2} + a)}{d x} = 2 x \\ \Rightarrow \frac{d f}{d x} = \frac{d f}{d U} \frac{d U}{d x} = (n U^{n - 1}) (2 x) = 2 n x {(x^{2} + a)}^{n - 1} \end{array}

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