3.3 differentiation rules - ds2: problem 3\n(6 points)\nif\n f(x)=\frac{6 x^{3}-5}{x^{4}} \nfind (…

3.3 differentiation rules - ds2: problem 3\n(6 points)\nif\n f(x)=\frac{6 x^{3}-5}{x^{4}} \nfind ( f^{prime}(x) ).\n( f^{prime}(x)= )\nfind ( f^{prime}(3) ).\n( f^{prime}(3)= )\nnote: you can earn partial credit on this problem.\nnote: you are in the reduced scoring period. all work counts for ( 85 % ) of the original.\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have 5 attempts remaining.

3.3 differentiation rules - ds2: problem 3\n(6 points)\nif\n f(x)=\frac{6 x^{3}-5}{x^{4}} \nfind ( f^{prime}(x) ).\n( f^{prime}(x)= )\nfind ( f^{prime}(3) ).\n( f^{prime}(3)= )\nnote: you can earn partial credit on this problem.\nnote: you are in the reduced scoring period. all work counts for ( 85 % ) of the original.\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have 5 attempts remaining.

Answer

Explanation:

Step1: Simplify the function

Rewrite ( f(x)=\frac{6x^{3}-5}{x^{4}} ) as ( f(x)=6x^{- 1}-5x^{-4} ).

Step2: Differentiate using power rule

The power rule is ( \frac{d}{dx}(x^{n})=nx^{n - 1} ). For ( y = 6x^{-1} ), ( y^\prime=6\times(-1)x^{-1 - 1}=-6x^{-2} ). For ( y=-5x^{-4} ), ( y^\prime=-5\times(-4)x^{-4 - 1}=20x^{-5} ). So ( f^\prime(x)=-6x^{-2}+20x^{-5}=\frac{-6}{x^{2}}+\frac{20}{x^{5}} ).

Step3: Find ( f^\prime(3) )

Substitute ( x = 3 ) into ( f^\prime(x) ). ( f^\prime(3)=\frac{-6}{3^{2}}+\frac{20}{3^{5}}=\frac{-6}{9}+\frac{20}{243} ). First, ( \frac{-6}{9}=-\frac{2}{3}=-\frac{162}{243} ). Then ( f^\prime(3)=-\frac{162}{243}+\frac{20}{243}=\frac{-162 + 20}{243}=\frac{-142}{243} ).

Answer:

( f^\prime(x)=\frac{-6}{x^{2}}+\frac{20}{x^{5}} ), ( f^\prime(3)=\frac{-142}{243} )