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Overview of Differentiation

April 9th, 2009 |

Author: Merrin

Mastering differentiation means knowing how to differentiate all the elementary functions as well as apply the differentiation rules to more complicated expressions involving the elementary functions. In this section, we list all the rules and derivatives without proof. In the rest of this chapter, we fill in the details and give examples on how to apply the rules.

After rereading this section, those who comprehend every formula should have a fairly good grasp of differentiation. For now, I suggest attempting to memorize all 30 or so rules. The definition of the derivative is used to prove all the differentiation rules and find some basic derivatives

$f'(x)=\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{f(x+\Delta x)-f(x)}{\Delta x}$

The derivative of a constant is zero

$\cfrac{d}{dx} c=0$

Differentiation is linear meaning that

$\cfrac{d}{dx} (c_{1} f(x)+c_{2} g(x))=c_{1} \cfrac{d}{dx} f(x)+c_{2} \cfrac{d}{dx} g(x)$

The power rule is for differentiating “powers laws” with real exponents

$\cfrac{d}{dx} x^{n} =nx^{n-1}$

The derivative of a product of functions is given by the product rule.

$\cfrac{d}{dx} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$

For the derivative of a composition of functions, use the chain rule.

$\cfrac{d}{dx} (f\circ g)=\cfrac{d}{dx} f(g(x))=\cfrac{df}{dg} \cfrac{dg}{dx}$

The reciprocal rule can be used to find the reciprocal of a function. Note that the reciprocal and the inverse are two separate things.

$\cfrac{d}{dx} \cfrac{1}{f(x)} =-\cfrac{1}{f^{2} (x)} \cfrac{df}{dx}$ The

quotient rule is the product rule where one of the functions is a reciprocal.

$\cfrac{d}{dx} \left(\cfrac{f(x)}{g(x)} \right)=\cfrac{f'(x)g(x)-f(x)g'(x)}{g^{2} (x)}$

Rational functions are the ratio of two polynomials and an example of functions that can be differentiated by the quotient rule. We can build up a catalog of derivatives by differentiating the elementary functions. Using the differentiation rules we can find the derivatives of the trigonometric functions.

$\cfrac{d}{dx} \sin x = +\cos x \qquad \cfrac{d}{dx} \cos x=-\sin x$

$\cfrac{d}{dx} \sec x = +\sec x\tan x \qquad \cfrac{d}{dx}\csc x=-\csc x\cot x$

$\cfrac{d}{dx} \tan x = +\sec ^2 x \qquad \cfrac{d}{dx} \cot x=-\csc ^{2} x$

The hyperbolic functions are defined analogously to the trigonometric functions and we can list their derivatives also.

$\cfrac{d}{dx} \sinh x = +\cosh x \qquad \cfrac{d}{dx} \cosh x=+\sinh x$

$\cfrac{d}{dx} \text{sech}\, x = -\text{sech}\, x\text{tanh}\, x \qquad \cfrac{d}{dx} \text{csch}\, x=-\text{csch}\, x\text{coth}\, x$

$\cfrac{d}{dx}\tanh x =+\text{sech}^2 x \qquad \cfrac{d}{dx}\text{coth}\,x=-\text{csch}^2 x$

The derivatives of exponential and logarithmic functions are also useful. The base e is natural and base a is arbitrary.

$\cfrac{d}{dx} e^{x} =e^{x} \qquad \qquad \cfrac{d}{dx} \ln x=\cfrac{1}{x}$

$\cfrac{d}{dx} a^{x} =a^{x} \ln a \qquad \qquad \cfrac{d}{dx} \log _{a} x = \cfrac{1}{x\ln a}$

Implicit differentiation can be used to differentiate both sides of a relation. This can be used to find relationships between rates of change or to find implicit derivatives in terms of both x and y rather than just x.

$\begin{array}{c} {g=h} \\ {\cfrac{d}{dx} g(x,y)=\cfrac{d}{dx} h(x,y)} \end{array}$

Taking a logarithm of both sides of an equation then using a property of the logarithms sometimes simplifies differentiation of products, quotients, and complicated exponentials. This is logarithmic differentiation.

$f'(x)=f(x)\cfrac{d}{dx} \ln f(x)$

There is a simple relationship between the derivatives of a function and its inverse.

$\begin{array}{c} {y=f(x)\quad x=f^{-1} (y)} \\ {\cfrac{dy}{dx} =\cfrac{1}{\cfrac{dx}{dy} } } \end{array}$

We can use the derivatives of inverses to find the derivatives of the remaining elementary functions inverse trigonometric functions and inverse hyperbolic functions.

$\cfrac{d}{dx} \text{arcsin}\, x = \cfrac{1}{\sqrt{1-x^{2}}}\qquad \cfrac{d}{dx} \text{arccos}\, x=-\cfrac{d}{dx}\text{arcsin}\, x$