Find {eq}\displaystyle f'(x) {/eq} for {eq}\displaystyle f(x)=\frac{\ln (1-x)}{e^{x}} {/eq}

Question:

Find {eq}\displaystyle f'(x) {/eq} for {eq}\displaystyle f(x)=\frac{\ln (1-x)}{e^{x}} {/eq}

Derivative using Quotient Rule of Differentiation:

The below differentiation formula is called the quotient rule, and it is used to determine a rational function's derivative.

{eq}\dfrac{d}{{dx}}\left[ {\dfrac{{f\left( x \right)}}{{g\left( x \right)}}} \right] = \dfrac{{g\left( x \right)f'\left( x \right) - f\left( x \right)g'\left( x \right)}}{{{g^2}\left( x \right)}} {/eq}

The below differentiation formulas are important to compute the derivatives of a function.

{eq}\begin{align*} \dfrac{d}{{dx}}\left( {{x^n}} \right) &= n{x^{n - 1}}&\left[ {{\rm{Power}}\;{\rm{Rule}}} \right]\\ \dfrac{d}{{dx}}f\left[ {g\left( x \right)} \right] &= f'\left[ {g\left( x \right)} \right] \cdot g'\left( x \right)&\left[ {{\rm{Chain}}\;{\rm{Rule}}} \right]\\ \dfrac{d}{{dx}}\left[ {f\left( x \right) \pm g\left( x \right)} \right] &= \dfrac{d}{{dx}}f\left( x \right) \pm \dfrac{d}{{dx}}g\left( x \right)&\left[ {{\rm{Sum}}\;{\rm{and}}\;{\rm{Difference}}\;{\rm{Rule}}} \right]\\ \dfrac{d}{{dx}}\left[ {c \cdot f\left( x \right)} \right] &= c\dfrac{d}{{dx}}f\left( x \right)&\left[ {{\rm{Constant}}\;{\rm{Multiple}}\;{\rm{Rule}}} \right] \end{align*} {/eq}

Answer and Explanation: 1

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Given Data:

  • The given rational function is: {eq}f\left( x \right) = \dfrac{{\ln \left( {1 - x} \right)}}{{{e^x}}} {/eq}

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When to Use the Quotient Rule for Differentiation

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Chapter 8 / Lesson 8
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The quotient rule can be used for differentiation when taking the derivative of a function divided by another function. Gain a better understanding of when to use the quotient rule and explore some examples in this lesson.


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