Questions: Differentiate. g(x) = 7^x / (5x + 2)

Solution Steps

To differentiate the function \( g(x) = \frac{7^x}{5x + 2} \), 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 is given by:

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

In this case, \( f(x) = 7^x \) and \( u(x) = 5x + 2 \). We will need to find the derivatives \( f'(x) \) and \( u'(x) \) separately. The derivative of \( 7^x \) is \( 7^x \ln(7) \), and the derivative of \( 5x + 2 \) is 5.

We start with the function given by

\[ g(x) = \frac{7^x}{5x + 2} \]

To differentiate \( g(x) \), we apply the quotient rule, which states:

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

where \( f(x) = 7^x \) and \( u(x) = 5x + 2 \).

We compute the derivatives:

• The derivative of \( f(x) = 7^x \) is

\[ f'(x) = 7^x \ln(7) \]

• The derivative of \( u(x) = 5x + 2 \) is

\[ u'(x) = 5 \]

Substituting \( f'(x) \), \( f(x) \), \( u'(x) \), and \( u(x) \) into the quotient rule gives:

\[ g'(x) = \frac{7^x (5x + 2) \ln(7) - 5 \cdot 7^x}{(5x + 2)^2} \]

We can simplify the expression for \( g'(x) \):

\[ g'(x) = \frac{7^x \left( (5x + 2) \ln(7) - 5 \right)}{(5x + 2)^2} \]

Final Answer