Video: Differentiating Rational Functions at a Point Using the Quotient Rule

Evaluate the rate of change of 𝑓(𝑥) = −(9/(4𝑥 − 7)) at 𝑥 = 4.

02:42

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.

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