Question

Use the reciprocal rule to determine the derivative of the function…

Answer 1

The function is given as,

`f(x)=(1)/(5-6x)`

Differentiating both sides with respect to 'x',

`f^(\prime)(x)=(d)/(dx)((1)/(5-6x))`

According to the reciprocal rule,

`(d)/(dx)((1)/(g(x)))=(-g^(\prime)(x))/(g(x)^2)`

Comparing with the expression,

`g(x)=5-6x`

The corresponding derivative is given by,

`\begin{gathered} g^(\prime)(x)=(d)/(dx)(5-6x) \\ g^(\prime)(x)=(d)/(dx)(5)-6\cdot(d)/(dx)(x) \\ g^(\prime)(x)=0-6(1) \\ g^(\prime)(x)=-6 \end{gathered}`

Substitute the values in the formula,

`\begin{gathered} (d)/(dx)((1)/(5-6x))=(-(-6))/((5-6x)^2) \\ (d)/(dx)((1)/(5-6x))=(6)/((5-6x)^2) \end{gathered}`

So the derivative of the given function becomes,

`f^(\prime)(x)=(6)/((5-6x)^2)`

Thus, the derivative of the given function is obtained as,

`f^(\prime)(x)=(6)/((5-6x)^2)`