Video: Evaluating an Expression Containing Derivatives of Rational Functions

Calculate π‘₯⁢(d𝑦/dπ‘₯), given that 𝑦 = (4π‘₯⁡ βˆ’ 5)/(8π‘₯⁡).

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Video Transcript

Calculate π‘₯ raised to the power of six times d𝑦 by dπ‘₯ given that 𝑦 is equal to four times π‘₯ raised to the power of five minus five all over eight times π‘₯ raised to the power of five.

We’re asked to calculate π‘₯ raised to the power of six times d𝑦 by dπ‘₯. And we’re given that 𝑦 is a rational function four π‘₯ raised to the power of five minus five over eight times π‘₯ raised to the power of five. In order to calculate this, we’re going to need to find the derivative of 𝑦 with respect to π‘₯. That’s d𝑦 by dπ‘₯. We could do this by dividing and then using the power rule for differentiation. However, we’re going to use the quotient rule to show how that works. And this says that if 𝑓 of π‘₯ is the ratio of 𝑒 of π‘₯ over 𝑣 of π‘₯, then the derivative of 𝑓 with respect to π‘₯, that’s 𝑓 prime of π‘₯, is equal to 𝑒 prime of π‘₯ times 𝑣 of π‘₯ minus 𝑒 of π‘₯ times 𝑣 prime of π‘₯ all over 𝑣 squared.

In our case, 𝑒 of π‘₯ is equal to four π‘₯ raised to the power five minus five. And 𝑣 of π‘₯ is equal to eight times π‘₯ raised to the power of five. And to find d𝑦 by dπ‘₯, we need to find 𝑒 prime of π‘₯ and 𝑣 prime of π‘₯. So with 𝑒 of π‘₯ equal to four π‘₯ raised to the power of five minus five, we can find d𝑒 by dπ‘₯ by using the power rule for differentiation. This says that if 𝑓 of π‘₯ is a function of the form π‘Ž times π‘₯ raised to the 𝑛th power, then 𝑓 prime of π‘₯, that’s d𝑓 by dπ‘₯, is equal to 𝑛 times π‘Ž times π‘₯ raised to the power of 𝑛 minus one. That is, we multiply by the exponent 𝑛 and then subtract one from the exponent.

In our first term, our exponent is five so that the derivative is five times four times π‘₯ raised to the power of four. And we know that the derivative of the constant negative five is zero so that 𝑒 prime of π‘₯ is 20 times π‘₯ raised to the power of four. Similarly, for 𝑣, our exponent is five again so that 𝑣 prime of π‘₯ is five times eight times π‘₯ raised to the power of four, that is, 40 times π‘₯ raised to the power of four.

So now let’s clear some space, and let’s use 𝑒, 𝑣, and their derivatives in the quotient rule to find d𝑦 by dπ‘₯. In our numerator, we have 𝑒 prime times 𝑣 minus 𝑒 times 𝑣 prime. And in our denominator, we have eight times π‘₯ raised to the power of five all squared, which is 𝑣 squared. In our numerator, we have a common factor of 20π‘₯ raised to the power of four. And we can take this outside of our bracket to get 20π‘₯ raised to the power of four times eight π‘₯ raised to the power of five minus two times four π‘₯ raised to the power of five minus five all over eight π‘₯ to the power of five squared.

And now, if we multiply out our internal bracket, in the numerator, we can see that the positive and negative eight π‘₯ to the power of five cancel each other out. So we have 10 times 20π‘₯ raised to the power of four over eight times π‘₯ raised to the power of five all squared. And that’s 200π‘₯ raised to the power of four times eight π‘₯ raised to the power of five all squared. And now, if we multiply out our denominator, we have d𝑦 by dπ‘₯ is equal to 200π‘₯ raised to the power of four over 64 times π‘₯ raised to the 10th power.

Now, if we refer back to our question, we’re actually asked for π‘₯ raised to the power of six times d𝑦 by dπ‘₯. And that’s equal to π‘₯ raised to the power of six times 200 times π‘₯ raised to the power of four over 64 times π‘₯ raised to the 10th power. And recalling that π‘₯ raised to the power of π‘Ž times π‘₯ raised to the power of 𝑏 is π‘₯ raised to the power of π‘Ž plus 𝑏, where in our case π‘Ž is six and 𝑏 is four, we have 200 times π‘₯ raised to the 10th power over 64 times π‘₯ raised to the 10th power. And so we can cancel our π‘₯ to the 10th powers and we’re left with 200 over 64. In its simplest form, that’s 25 over eight.

And so, given 𝑦 equal to four times π‘₯ raised to the power of five minus five divided by eight times π‘₯ to the power of five, then π‘₯ raised to the power of six times d𝑦 by dπ‘₯ is equal to 25 over eight.

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