9–61. Evaluate and simplify y'.

y = 5t² ...

Question

9–61. Evaluate and simplify y'.

y = 5t² sin t

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says determine the derivative of Z with respect to W, given that Z is equal to 2, W4 multiplied by tangent of W. We're given 4 possible choices as our answers. For choice A, we have Z of W is equal to 2 W cubed, multiplying the quantity of 4 multiplied by tangent of W, plus W multiplied by c2d of W. For choice B, we have Z W is equal to 2 W q multiplied by the quantity. Of 4 multiplied by tangent of W, minus W multiplied by square of W. For choice C we have Z W is equal to 2 W cubed, multiplying the quantity of 4 multiplied by squad of W, plus W multiplied by tangent of W. And for choice D we have Z W is equal to 2 W cubed, multiplied by the quantity of 4 multiplied by c2 W minus W multiplied by tangent of W. Now we're asked to find the derivative of Z with respect to W, and if I look at our function of Z, we're going to need to apply our product rule in order to take this derivative. So, we call for the product rule that if I try to take the derivative of the product of F multiplied by G. Then this is going to be equal to F multiplied by G plus F multiplied by G. Now if we look at our function Z, we can identify F as being equal to 2 multiplied by W to the fourth. So that means that F is going to be equal to 8 multiplied by W cubed, and here we just applied our power rule, and that means that our function G is going to be equal to the tangent of W. And that means that G is going to be equal to squad of W. So now we have all the information that we need to calculate our derivative of Z with respect to W, and so that's gonna be Z of W. And that is going to be equal to, and here we'll apply our product rule. So we'll have F which is 8 multiplied by W cubed, and that's going to be multiplied by G, which is tangent of W. Then we'll have plus our function F, which is 2 multiplied by W to the fourth, multiplied by G which is squared of W. I noticed that I can factor out a 2 W cubed out of each term, so that means that this is going to be equal to 2 W cubed. Multiplying the quantity of 4, multiplied by the tangent of W. Plus, W multiplied by the squared of W. So this is the answer that we're actually looking for. And if we compare our answer that we've calculated to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

9–61. Evaluate and simplify y'.

y = 5t² sin t

Key Concepts

Differentiation

Product Rule

Trigonometric Functions

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