# How do you use the product rule to find the derivative of y=(1/x^2-3/x^4)*(x+5x^3) ?

Jul 25, 2014

$y ' = \left(\frac{1}{x} ^ 2 - \frac{3}{x} ^ 4\right) \cdot \left(1 + 15 {x}^{2}\right) + \left(- \frac{2}{x} ^ 3 + \frac{12}{x} ^ 5\right) \left(x + 5 {x}^{3}\right)$

Solution :
Suppose we have $y = f \left(x\right) \cdot g \left(x\right)$
Then, using Product Rule, $y ' = f \left(x\right) \cdot g ' \left(x\right) + f ' \left(x\right) \cdot g \left(x\right)$

In simple language, keep first term as it is and differentiate the second term, then differentiate the first term and keep the second term as it is or vice-versa.

So, here if we consider,

$f \left(x\right) = \left(\frac{1}{x} ^ 2 - \frac{3}{x} ^ 4\right)$
$g \left(x\right) = \left(x + 5 {x}^{3}\right)$

Then,

$f ' \left(x\right) = \left(- \frac{2}{x} ^ 3 + \frac{12}{x} ^ 5\right)$
$g ' \left(x\right) = \left(1 + 15 {x}^{2}\right)$

Hence, using the product rule,

$y ' = \left(\frac{1}{x} ^ 2 - \frac{3}{x} ^ 4\right) \left(1 + 15 {x}^{2}\right) + \left(- \frac{2}{x} ^ 3 + \frac{12}{x} ^ 5\right) \left(x + 5 {x}^{3}\right)$

In case , if we have more than two function, let see

$y = u \left(x\right) \cdot v \left(x\right) \cdot w \left(x\right)$

then,

$y ' = u ' \left(x\right) \cdot v \left(x\right) \cdot w \left(x\right) + u \left(x\right) \cdot v ' \left(x\right) \cdot w \left(x\right) + u \left(x\right) \cdot v \left(x\right) \cdot w ' \left(x\right)$

i.e. differentiate one function at a time and keep the remaining two as it is or consider them as constant and similarly follow for the remaining two.