Question Video: Finding the Second Derivative of a Polynomial Function Mathematics • Higher Education

Given that ๐‘ฆ = 6๐‘ฅโต + 3๐‘ฅยฒ โˆ’ 7๐‘ฅ + 6, determine dยฒ๐‘ฆ/d๐‘ฅยฒ.

02:25

Video Transcript

Given that ๐‘ฆ is equal to six ๐‘ฅ to the power of five plus three ๐‘ฅ squared minus seven ๐‘ฅ plus six, determine the second derivative of ๐‘ฆ with respect to ๐‘ฅ.

Here weโ€™ve been given a function in ๐‘ฅ. And weโ€™re being asked to find d two ๐‘ฆ by d๐‘ฅ squared. To do this, weโ€™ll differentiate once to find d๐‘ฆ by d๐‘ฅ then differentiate that once again to find the second derivative. We recall that we can differentiate a function of the form ๐‘Ž๐‘ฅ to the power of ๐‘› with respect to ๐‘ฅ for some constant rational number ๐‘›, which is not equal to zero and some constant ๐‘Ž. And we get ๐‘›๐‘Ž times ๐‘ฅ to the power of ๐‘› minus one. In other words, we borrow the power of ๐‘ฅ and we make it the coefficient of the derivative. And then, we subtract one from the power.

In this special case where ๐‘› is equal to zero, we actually have a constant. Letโ€™s call that ๐‘. And the derivative of a constant is zero. This is going to help us find the first derivative of our function. The derivative of six ๐‘ฅ to the power of five is going to be five times six ๐‘ฅ. And then, we subtract one from the power. Five minus one is four. Thatโ€™s 30๐‘ฅ to the fourth power. Weโ€™ll repeat this to differentiate three ๐‘ฅ squared. Itโ€™s going to be two times three ๐‘ฅ. And then, we subtract one from the power. Two minus one is one. So, the derivative of three ๐‘ฅ squared with respect to ๐‘ฅ is six ๐‘ฅ.

The derivative of negative seven ๐‘ฅ is one times negative seven ๐‘ฅ to the power of zero. Well, thatโ€™s just negative seven. And, of course, six is a constant, so the derivative of six is zero. d๐‘ฆ by d๐‘ฅ then, the first derivative of our equation, is 30๐‘ฅ to the power of four plus six ๐‘ฅ minus seven. Weโ€™ll differentiate each part of this expression once more to find the second derivative.

Weโ€™ll do it piece-by-piece. The derivative of 30๐‘ฅ to the power of four is four times 30๐‘ฅ to the power of three. The derivative of six ๐‘ฅ is six. And the derivative of negative seven is zero. Simplifying fully, and we see that the second derivative of ๐‘ฆ with respect to ๐‘ฅ is 120๐‘ฅ cubed plus six.

In this example, we saw how we could apply repeated differentiation to help us find the second derivative of a polynomial function.

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