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Question Video: Finding the 𝑛th Derivative of a Power Function Where Its Exponent Is 𝑛 + 1 Mathematics • Higher Education

If 𝑦 = π‘₯⁹, find d⁸𝑦/dπ‘₯⁸.

03:02

Video Transcript

If 𝑦 is equal to π‘₯ to the ninth power, find the eighth derivative of 𝑦 with respect to π‘₯.

We’re given that 𝑦 is equal to π‘₯ to the ninth power, and we’re asked to find the eighth derivative of 𝑦 with respect to π‘₯. This means we’re going to need to differentiate our expression for 𝑦 eight times. To do this, we’re going to need to start by recalling the power rule for differentiation. This tells us for any real constants π‘Ž and 𝑛, the derivative of π‘Žπ‘₯ to the 𝑛th power with respect to π‘₯ is equal to π‘Ž times 𝑛 times π‘₯ to the power of 𝑛 minus one. We multiply by our exponent of π‘₯ and then reduce this exponent by one.

We can start by using this to find the first derivative of 𝑦 with respect to π‘₯. So by differentiating our expression for 𝑦, we get d𝑦 by dπ‘₯ will be equal to the derivative of π‘₯ to the ninth power with respect to π‘₯. And we can do this by using the power rule for differentiation. We set our exponent value of 𝑛 equal to nine and our value of π‘Ž equal to one. So we multiply by our exponent of nine and reduce this exponent by one. This gives us nine π‘₯ to the eighth power.

We can then find an expression for the second derivative of 𝑦 with respect to π‘₯ by differentiating both sides of this equation with respect to π‘₯. We get d two 𝑦 by dπ‘₯ squared is equal to the derivative of nine π‘₯ to the eighth power with respect to π‘₯. Once again, we can differentiate this by using the power rule for differentiation. This time, our value of 𝑛 will be equal to eight and our value of π‘Ž will be equal to nine. So we need to multiply by our exponent of eight and reduce this exponent by one. This gives us nine times eight π‘₯ to the seventh power. And we can write this as nine times eight times π‘₯ to the seventh power.

And once again, we can see we can differentiate this by using the power rule for differentiation. And we’re starting to see a pattern. If we were to differentiate this by using the power rule for differentiation, we would multiply by our exponent of seven and then reduce this exponent by one. And if every time we differentiate our expression we’re reducing our exponent by one, this means we’re reducing our exponent by the order of our derivative. For example, in our third derivative of 𝑦 with respect to π‘₯, we’ve differentiated π‘₯ to the ninth power with respect to π‘₯ three times. So we need to reduce this exponent by three. And of course we’ll also multiply by each of our exponents in turn. In this case, it will be nine times eight times seven. Remember, we need to reduce our exponent by one each time.

We can use this exact same reasoning to find an expression for the eighth derivative of 𝑦 with respect to π‘₯. First, we know we’ll get a coefficient which is eight of our exponents all multiplied together. And each time we reduce our exponent by one. We also know this starts with nine. This then gives us a coefficient of nine times eight times seven, and we multiply all the way down to two. And of course, we can simplify this. First, multiplying by one doesn’t change our value. And now we can see this is exactly equal to nine factorial.

Next, remember, every time we differentiate π‘₯ to the ninth power, we reduce its exponent by one. We see we’ve taken the eighth derivative of 𝑦 with respect to π‘₯, so we need to reduce this exponent by eight. So we need to multiply this coefficient by π‘₯ to the power of nine minus eight. And of course, we can simplify this. Nine minus eight is equal to one. And π‘₯ to the first power is just equal to π‘₯. So by reordering these two factors, we get our final answer of π‘₯ times nine factorial.

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