The function is given as,
[tex]f(x)=\frac{1}{5-6x}[/tex]
Differentiating both sides with respect to 'x',
[tex]f^{\prime}(x)=\frac{d}{dx}(\frac{1}{5-6x})[/tex]
According to the reciprocal rule,
[tex]\frac{d}{dx}(\frac{1}{g(x)})=\frac{-g^{\prime}(x)}{g(x)^2}[/tex]
Comparing with the expression,
[tex]g(x)=5-6x[/tex]
The corresponding derivative is given by,
[tex]\begin{gathered} g^{\prime}(x)=\frac{d}{dx}(5-6x) \\ g^{\prime}(x)=\frac{d}{dx}(5)-6\cdot\frac{d}{dx}(x) \\ g^{\prime}(x)=0-6(1) \\ g^{\prime}(x)=-6 \end{gathered}[/tex]
Substitute the values in the formula,
[tex]\begin{gathered} \frac{d}{dx}(\frac{1}{5-6x})=\frac{-(-6)}{(5-6x)^2} \\ \frac{d}{dx}(\frac{1}{5-6x})=\frac{6}{(5-6x)^2} \end{gathered}[/tex]
So the derivative of the given function becomes,
[tex]f^{\prime}(x)=\frac{6}{(5-6x)^2}[/tex]
Thus, the derivative of the given function is obtained as,
[tex]f^{\prime}(x)=\frac{6}{(5-6x)^2}[/tex]
