Video: Finding the Second Derivative of a Polynomial Function

Video Transcript

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

Here we’ve been given a function in 𝑥. And we’re being asked to find d two 𝑦 by d𝑥 squared. To do this, we’ll differentiate once to find d𝑦 by d𝑥 then differentiate that once again to find the second derivative. We recall that we can differentiate a function of the form 𝑎𝑥 to the power of 𝑛 with respect to 𝑥 for some constant rational number 𝑛, which is not equal to zero and some constant 𝑎. And we get 𝑛𝑎 times 𝑥 to the power of 𝑛 minus one. In other words, we borrow the power of 𝑥 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 𝑥 to the power of five is going to be five times six 𝑥. And then, we subtract one from the power. Five minus one is four. That’s 30𝑥 to the fourth power. We’ll repeat this to differentiate three 𝑥 squared. It’s going to be two times three 𝑥. And then, we subtract one from the power. Two minus one is one. So, the derivative of three 𝑥 squared with respect to 𝑥 is six 𝑥.

The derivative of negative seven 𝑥 is one times negative seven 𝑥 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 d𝑥 then, the first derivative of our equation, is 30𝑥 to the power of four plus six 𝑥 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 30𝑥 to the power of four is four times 30𝑥 to the power of three. The derivative of six 𝑥 is six. And the derivative of negative seven is zero. Simplifying fully, and we see that the second derivative of 𝑦 with respect to 𝑥 is 120𝑥 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.