Video: Finding the Value of the Second Derivative of a Function at a Given Point

Determine the value of the second derivative of the function 𝑦 = 12π‘₯ βˆ’ (8/π‘₯) at (1, 4).

01:50

Video Transcript

Determine the value of the second derivative of the function 𝑦 equals 12π‘₯ minus eight over π‘₯ at one, four.

We have an equation for 𝑦 in terms of π‘₯. And we’re being asked to find the value of the second derivative at the point with Cartesian coordinates one, four. We’ll begin then by simply finding an expression for the second derivative. To do this, we’ll differentiate our function once to find d𝑦 by dπ‘₯ then differentiate with respect to π‘₯ once more. It might help, before we do, to write 𝑦 as 12π‘₯ minus eight π‘₯ to the negative one. Then, we differentiate as normal.

The derivative of 12π‘₯ with respect to π‘₯ is simply 12. And the derivative of negative eight π‘₯ to the power of negative one is negative negative one times eight π‘₯ to the power of negative two. And be really careful. A common mistake here is to spot the negative one and think that when we subtract one, we get to zero. When we simplify this, we see that the first derivative is 12 plus eight π‘₯ to the negative two. Let’s repeat this process to find the second derivative.

The derivative of 12 is zero. And then, when we differentiate eight π‘₯ to the negative two with respect to π‘₯, we get negative two times eight π‘₯ to the negative three. That’s negative 16π‘₯ to the negative three. And, of course, we can change this back to negative 16 over π‘₯ cubed if we like.

We need to determine the value of the second derivative at one, four. This is a Cartesian coordinate. It has an π‘₯-value of one and a 𝑦-value of four. So, we’re going to substitute π‘₯ is equal to one into our equation for the second derivative. This gives us negative 16 over one cubed, which is negative 16.

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