Video: Finding the First Derivative of a Function Using the Power Rule

Given that 𝑦 = 15 − (1/𝑥⁶) + (1/3)𝑥²⁰, determine 𝑦′.

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Video Transcript

Given that 𝑦 equals 15 minus one over 𝑥 to the power of six plus a third 𝑥 to the power of 20, determine the derivative of 𝑦.

So the first stage is to actually to rewrite our function of 𝑦. So we’re gonna rewrite it as 𝑦 equals 15 minus 𝑥 to the power of negative six — and we get that because we’ve got the general rule that says if you have 𝑥 to the power of negative 𝑎, then this is equal to one over 𝑥 to the power of 𝑎 — and then plus a third 𝑥 to the power of 20. So our final term doesn’t change.

Okay, now we’ve done that, it means it should be easier to differentiate. So now, if we actually differentiate our function, the first term is gonna be zero. And that’s because if you actually differentiate any term without 𝑥, we just get zero and then we get plus six 𝑥 to the power of negative seven. So just to remind ourselves how we actually got that, we’ve got negative one multiplied by negative six and that’s because that’s our coefficient multiplied by our exponent, which just gives us positive six.

And then, we had 𝑥 to the power of negative six minus one cause you subtract one from the exponent which gives us negative seven. And then, we get plus 20 over three 𝑥 to the power of 19. And we got this final term because again we multiplied our coefficient, which was a third, by our exponent, which was 20. So that gave us 20 over three. And then, we reduce the exponent of our 𝑥 term from 20 to 19 because we subtracted one. So we got 20 over three 𝑥 to the power of 19.

Okay, so next, we can actually tidy this up. So therefore, we get that the derivative of 𝑦 equals 15 minus one over 𝑥 to the power of six plus a third 𝑥 to the power of 20 is equal to six over 𝑥 to the power of seven plus 20 over three 𝑥 to the power of 19.

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