rationalskepticism.org

Posted: **Feb 24, 2012 9:03 am**

QUOTIENT RULE

The quotient rule is about two functions that are divided with each other: so given two functions, f(x) and g(x) that are being divided, how do we differentiate?

h(x) = \frac {f(x)}{g(x)}

The derivative will be:

h'(x) = \frac {f'(x)g(x) - f(x)g'(x)}{g^2(x)}

Using words: the derivative of two functions being divided is given by the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator; all this divided by the square of the denominator. Sounds complicated right? However, once you practice it a bit, it will be very easy to do. Let's see a simple example:

h(x)=\frac {\sin{x}}{x^2}

In the above we have:

f(x) = \sin {x}

g(x) = x^2

When we differentiate the above two we get:

f'(x) = \cos {x}

g'(x) = 2x

How about the square of the denominator?

g^2(x) = (x^2)^2

g^2(x) = x^4

Then let us apply the rule as above:

h'(x)=\frac {x^2\cos{x} - 2x\sin {x}}{x^4}

The quotient rule takes some practice. But it is not as difficult as it may seem at first glance. Tomorrow I will finish up with the composite rule; I wanted to do it today, but explaining the quotient rule I saw that it may be better to do the last rule tomorrow, and let you read and understand the quotient rule first.

rationalskepticism.org

Calculus: Differentiation Made Easy