How do you find the derivative for #f(x)=(18x)/(4+(x^2))#?

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Answer 1 · 2018-05-09T21:35:26

#=(-18(x^2-4))/(x^2+4)^2#

We're dealing with a rational function here, so we can use the quotient rule stated below:

#f'(x)=(g'(x)h(x)-g(x)h'(x))/(h(x)^2)#

If we define #color(blue)(g(x)=18x)# and #color(purple)(h(x)=x^2+4)#, we know

#color(blue)(g'(x)=18)# and #color(purple)(h'(x)=2x)#

Now we can plug in! We get

#f'(x)=(color(blue)(18)color(purple)((x^2+4))-color(blue)(18x)color(purple)((2x)))/(x^2+4)^2#

#=(-36x^2+18x^2+72)/(x^2+4)^2#

#=(-18x^2+72)/((x^2+4)^2#

#=(-18(x^2-4))/(x^2+4)^2#

Hope this helps!