How do you find the derivative of #6(z^2+z-1)^-1#?

2 Answers
Aug 30, 2017

#-(12z+6)/(z^2+z-1)^2#

Explanation:

#"differentiate using the "color(blue)"chain rule"#

#"given "y=f(g(x)" then"#

#dy/dx=f'(g(x))xxg'(x)larr" chain rule"#

#d/dz(6(z^2+z-1)^-1)#

#=-6(z^2+z-1)^-2xxd/dz(z^2+z-1)#

#=(-6(2z+1))/(z^2+z-1)^2=-(12z+6)/(z^2+z-1)^2#

Aug 30, 2017

Recall the power rule: #d/dxx^n=nx^(n-1)#. When we have a function to a power, we still use the power rule to differentiate it, but we do so as well as using the chain rule.

Combining the chain rule with the power rule for some function #u# gives us: #d/dxu^n=n u^(n-1)(du)/dx#

Thus:

#d/(dz)6(z^2+z-1)^-1=6(-1(z^2+z-1)^-2)d/(dz)(z^2+z-1)#

And we can use the power rule to find the derivative of #z^2+z-1#:

#d/(dz)6(z^2+z-1)^-1=-6(z^2+z-1)^-2(2z+1)#

#=color(blue)((-6(2z+1))/(z^2+z-1)^2#