How do you use the quotient rule to differentiate #f(x) = (x^1.7 + 2) / (x^2.8 + 1)#?

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Answer 1 · 2016-03-06T19:24:29

#f'(x)=(1.7x^0.7-5.6x^1.8-1.1x^3.5)/(x^2.8+1)^2#

The quotient rule states that

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

In the given function, we see that

#g(x)=x^1.7+2#
#h(x)=x^2.8+1#

To find these functions' derivatives, use the power rule, which states that

#d/dx(x^n)=nx^(n-1)#

Hence:

#g'(x)=1.7x^(1.7-1)=1.7x^0.7#
#h'(x)=2.8x^(2.8-1)=2.8x^1.8#

Plugging these functions into the quotient rule expression, we see that

#f'(x)=((x^2.8+1)(1.7x^0.7)-(x^1.7+2)(2.8x^1.8))/(x^2.8+1)^2#

When simplifying, recall that #x^a(x^b)=x^(a+b)#.

#f'(x)=(1.7x^3.5+1.7x^0.7-2.8x^3.5-5.6x^1.8)/(x^2.8+1)^2#

#f'(x)=(1.7x^0.7-5.6x^1.8-1.1x^3.5)/(x^2.8+1)^2#