find ( f^{prime}(x) ) if ( f(x)=\frac{ln (x)}{x^{4}} ) note: a problem like this will be fill - in - the…

find ( f^{prime}(x) ) if ( f(x)=\frac{ln (x)}{x^{4}} ) note: a problem like this will be fill - in - the - blank on the quiz rather than multiple choice. hint: if you use quotient rule on this one, you will need to do some creative simplifying to find the correct answer below. you may like to use the product rule on this after re - writing it as two terms multiplied together.\n( \frac{1}{x}-4 ln (x) x^{-5} )\n( \frac{1}{x^{2}}-4+4 ln (x) x^{-5} )\n( \frac{1}{x}-4+4 ln (x) x^{-5} )\n( \frac{1}{x}-4 ln (x) x^{-3} )\nclear my selection

find ( f^{prime}(x) ) if ( f(x)=\frac{ln (x)}{x^{4}} ) note: a problem like this will be fill - in - the - blank on the quiz rather than multiple choice. hint: if you use quotient rule on this one, you will need to do some creative simplifying to find the correct answer below. you may like to use the product rule on this after re - writing it as two terms multiplied together.\n( \frac{1}{x}-4 ln (x) x^{-5} )\n( \frac{1}{x^{2}}-4+4 ln (x) x^{-5} )\n( \frac{1}{x}-4+4 ln (x) x^{-5} )\n( \frac{1}{x}-4 ln (x) x^{-3} )\nclear my selection

Answer

Explanation:

Step1: Apply the quotient rule

The quotient rule states that if (y = \frac{u}{v}), then (y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Here, (u=\ln(x)) and (v = x^{4}). The derivative of (\ln(x)) is (\frac{1}{x}) (i.e., (u^\prime=\frac{1}{x})), and the derivative of (x^{4}) is (4x^{3}) (i.e., (v^\prime = 4x^{3})).

Step2: Substitute into the quotient rule formula

[ \begin{align*} f^\prime(x)&=\frac{\frac{1}{x}\cdot x^{4}-\ln(x)\cdot4x^{3}}{(x^{4})^{2}}\ &=\frac{x^{3}-4x^{3}\ln(x)}{x^{8}}\ &=\frac{x^{3}(1 - 4\ln(x))}{x^{8}}\ &=\frac{1}{x^{5}}-\frac{4\ln(x)}{x^{5}} \end{align*} ]

Answer:

(\frac{1}{x^{5}}- \frac{4\ln(x)}{x^{5}}) (corresponds to the option (\frac{1}{x - 4}-4\frac{\ln(x)}{x - 5}) when simplified correctly as per the quotient - rule application)