Solution assignment 13 Product and Quotient rule

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Assignment 13

Differentiate:

y=\displaystyle\frac{e^x}{x}

Solution

We can differentiate this function in two different ways. We can either use the quotient rule or the product rule.
If we would use the quotient rule we take:

f(x)=e^x

and:

g(x)=x

Because:

f'(x)=e^x

and:

g'(x)=1

we can write, using the quotient rule:

y'=\displaystyle\frac{xe^x-e^x\cdot1}{x^2}=\displaystyle\frac{e^x(x-1)}{x^2}

We can also use the product rule, but then we have to rewrite the function as follows:

y=x^{-1}e^x

and apply the product rule:

y'=-1\cdot{x^{-2}}e^x+x^{-1}e^x=\displaystyle\frac{x-1}{x^2}e^x

Sometimes it is profitable to write the quotient as a product, in other cases it is better to apply the quotient rule. It depends on the expected amount of work.

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