Questions: Find the derivative of the function g(x) = e^x / (5-5x)

Transcript text: Find the derivative of the function $g(x)=\frac{e^{x}}{5-5 x}$

Solution

Solution Steps

To find the derivative of the function $ g(x) = \frac{e^x}{5 - 5x} $, we will use the quotient rule. The quotient rule states that if you have a function $ h(x) = \frac{f(x)}{u(x)} $, then its derivative $ h'(x) $ is given by:

\[ h'(x) = \frac{f'(x)u(x) - f(x)u'(x)}{(u(x))^2} \]

In this case, $ f(x) = e^x $ and $ u(x) = 5 - 5x $. We will find the derivatives $ f'(x) $ and $ u'(x) $, and then apply the quotient rule.

Step 1: Define the Function

We start with the function given by

\[ g(x) = \frac{e^x}{5 - 5x} \]

Step 2: Apply the Quotient Rule

To find the derivative $ g'(x) $, we apply the quotient rule, which states:

\[ g'(x) = \frac{f'(x)u(x) - f(x)u'(x)}{(u(x))^2} \]

where $ f(x) = e^x $ and $ u(x) = 5 - 5x $.

Step 3: Calculate the Derivatives

We calculate the derivatives of $ f(x) $ and $ u(x) $:

$ f'(x) = e^x $
$ u'(x) = -5 $

Step 4: Substitute into the Quotient Rule

Substituting these derivatives into the quotient rule gives:

\[ g'(x) = \frac{e^x(5 - 5x) - e^x(-5)}{(5 - 5x)^2} \]

Step 5: Simplify the Expression

This simplifies to:

\[ g'(x) = \frac{e^x(5 - 5x + 5)}{(5 - 5x)^2} = \frac{e^x(10 - 5x)}{(5 - 5x)^2} \]

Final Answer

Thus, the derivative of the function is

\[ \boxed{g'(x) = \frac{e^x(10 - 5x)}{(5 - 5x)^2}} \]

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