### Video Transcript

Evaluate the rate of change of 𝑓
of 𝑥 equals negative nine divided by four 𝑥 minus seven at 𝑥 equals four.

The rate of change can be
calculated by differentiating to work out 𝑑𝑦 by 𝑑𝑥 or 𝑓 dash of 𝑥. In our example, the function is
equal to negative nine divided by four 𝑥 minus seven. One divided by 𝑥 is the same as 𝑥
to the power of negative one. This means in our example, 𝑓 of 𝑥
can be rewritten negative nine multiplied by four 𝑥 minus seven to the power of
negative one.

We now need to differentiate this
function to work out 𝑓 dash of 𝑥. Multiplying the exponent negative
one by negative nine gives us positive nine. Decreasing the power gives us
negative two — four 𝑥 minus seven to the power of negative two. We then need to multiply this by
the parenthesis differentiated. In this case, the differential of
four 𝑥 minus seven is equal to four.

Simplifying this gives us 𝑓 dash
of 𝑥 is equal to 36 multiplied by four 𝑥 minus seven to the power of negative two,
which in turn can be rewritten as 36 divided by four 𝑥 minus seven squared.

We were asked to work out the rate
of change at 𝑥 equals four. Therefore, we need to work out 𝑓
dash of four. Substituting in 𝑥 equals four
gives us 36 divided by four multiplied by four minus seven all squared. This gives us an answer of 36 over
81 — 36 divided by 81. We can simplify this fraction by
dividing the numerator and the denominator by nine. 36 divided by nine is equal to four
and 81 divided by nine is equal to nine.

This means that the rate of change
of 𝑓 of 𝑥 equals negative nine divided by four 𝑥 minus seven at 𝑥 equals four is
equal to four ninths. This tells us that the slope or
gradient of a function at 𝑥 equals four is four ninths.