How do you differentiate #f(x)=sqrt(x-(3x+5)^2)# using the chain rule.?

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Answer 1 · 2016-01-22T17:41:35

#f'(x)=-(18x+29)/(2sqrt(-9x^2-29x+25))#

This will be simpler if the terms inside the square root are simplified.

#f(x)=sqrt(x-(9x^2+30x+25))=sqrt(-9x^2-29x+25)#

Now, to differentiate a square root function, we should treat it as having a fractional exponent.

#f(x)=(-9x^2-29x+25)^(1/2)#

According to the chain rule, which we will have to use, #d/dx[u^(1/2)]=1/2u^(-1/2)*u'#, and we have #u=-9x^2-29x+25#.

Thus,

#f'(x)=1/2(-9x^2-29x+25)^(-1/2)*d/dx[-9x^2-29x+25]#

#f'(x)=1/(2(-9x^2-29x+25)^(1/2))*(-18x-29)#

#f'(x)=-(18x+29)/(2sqrt(-9x^2-29x+25))#