How do you differentiate #g(z)=(z^2+1)/(x^3-5)# using the quotient rule?

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Answer 1 · 2017-09-20T21:12:26

#(-z^4 - 3z^2 -10z)/(z^6 - 10z^3 +25)#

The quotient rule can be stated as;

#d/dz f(z)/g(z) = (f'(z)g(z) - f(z)g'(z))/g^2(z)#

We can choose our #f(z)# and #g(z)# and take each derivative separately. We only need the power rule here.

#f(z) = z^2+1#
#f'(z) = 2z#

#g(z) = z^3-5#
#g'(z) = 3z^2#

Now we have all of the pieces we need, we can plug them into our power rule function.

#(2z(z^3-5) - 3z^2(z^2+1))/(z^3-5)^2#

Now we can simplify our terms.

#(2z^4 - 10z - 3z^4 - 3z^2)/(z^6-10z^3 +25)#

#(-z^4 - 3z^2 -10z)/(z^6 - 10z^3 +25)#