Video: Finding the Second Derivative of a Polynomial Function

Given that 𝑦 = 6π‘₯⁡ + 3π‘₯Β² βˆ’ 7π‘₯ + 6, determine d²𝑦/dπ‘₯Β².

02:25

Video Transcript

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

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

In this special case where 𝑛 is equal to zero, we actually have a constant. Let’s call that 𝑏. 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 π‘₯ to the power of five is going to be five times six π‘₯. And then, we subtract one from the power. Five minus one is four. That’s 30π‘₯ to the fourth power. We’ll repeat this to differentiate three π‘₯ squared. It’s going to be two times three π‘₯. And then, we subtract one from the power. Two minus one is one. So, the derivative of three π‘₯ squared with respect to π‘₯ is six π‘₯.

The derivative of negative seven π‘₯ is one times negative seven π‘₯ to the power of zero. Well, that’s just negative seven. And, of course, six is a constant, so the derivative of six is zero. d𝑦 by dπ‘₯ then, the first derivative of our equation, is 30π‘₯ to the power of four plus six π‘₯ minus seven. We’ll differentiate each part of this expression once more to find the second derivative.

We’ll do it piece-by-piece. The derivative of 30π‘₯ to the power of four is four times 30π‘₯ to the power of three. The derivative of six π‘₯ is six. And the derivative of negative seven is zero. Simplifying fully, and we see that the second derivative of 𝑦 with respect to π‘₯ is 120π‘₯ 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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.