Solution assignment 14 Product and Quotient rule

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Assignment 14

Differentiate:

y=\tan(x)

Solution

This function is not in the list of standard functions. However, from trigonometry we know:

\tan(x)=\displaystyle\frac{\sin(x)}{\cos)x)}

and thus we can apply the quotient rule with:

f(x)=\sin(x)

and:

g(x)=\cos(x)

and thus:

f'(x)=\cos(x)

and:

g'(x)=\sin(x)

and thus the quotient rule yields:

y'=\displaystyle\frac{\cos(x)\cos(x)-\sin(x)\cdot-\sin(x)}{\cos^2(x)}=\displaystyle\frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)}

We can simplify this formula in two different ways. Because:

\cos^2(x)+\sin^2(x)=1

we can write:

y'=\displaystyle\frac{1}{\cos^2(x)}

Or, when we rewrite the original function we get the following form:

y'=\displaystyle\frac{\cos^2(x)}{\cos^2(x)}+\displaystyle\frac{\sin^2(x)}{\cos^2(x)}=1+(\displaystyle\frac{\sin(x)}{\cos(x)})^2=1+\tan^2(x)

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