Find the value of dy/dt at t = 0 if y = 3 sin 2x and x = t² + ...

Question

Find the value of dy/dt at t = 0 if y = 3 sin 2x and x = t² + π.

Accepted Answer

Welcome back, everyone. If Y equals 4 cosine of 3 X and X equals t cubed minus 5, what is the value of D Y divided by D T at T equals 0? We're given 4 answer choices A says -2, B -1, C0, and D1. So first of all, let's notice that Y. is a function of X and X is a function of t. And we're looking for the derivative of the y divided by D T, which is simply derivative of Y with respect to T, right? So this means that since we have a composite function, we will need to apply the chain rule which says that we are first rule differentiating Y with respect to X and multiplying the result by the derivative of X with respect to T. This means that we want to identify two derivatives. Let's begin with D Y divided by the X. So we went to differentiate 4, cosine of 3 X. Now we can factor out the constant we get for the derivative of cosine is negative sign, so we get negative sign of the original angle 3 X. But then because we have an inner function 3 X, once again we are applying the chain rule. So we're multiplying by the derivative of the inner function, which is the derivative of 3 X, and the derivative of 3 X is 3. So we get 3 multiplied by 4, that's 12, and since we have a negative sign, we're getting -12 of 3 X. Next, we want to evaluate D X divided by the C, the derivative of X with respect to T. And this means that we're differentiating T cubed minus pi. So let's differentiate T cubed. We can apply the power rule which gives us 3 T squad, and the derivative of negative pi is 0 because it's a constant, right? And that's it. We can now say that D Y divided by DT is simply the product between these two results. So we take -12 sin. Of 3 Xs multiplied by 3 T2. Combining the like terms, we can see that we have negative 36 T 2 sin. Of 3 X and now we want to evaluate the y divided by the T at T equals 0. So this means we simply want to substitute t equals 0, but we don't have T, right? We only have X. So what we're going to do is simply solve for X. X at 0.0, so this would be 0 cubed because we have T cubed, right, minus 5. So 0 cub minus pi gives us negative pi. So if T is 0, that means X is going to be equal to negative pi. So we're going to get -36. Multiplied by T2. So in this context, T2d is just 0 squared, but now sign of 3 X. Well, it essentially gives us Sign of negative pi. And to be extremely accurate, we can also say that the Y divided by DT can be expressed in terms of T only in order not to have that X right? so we can essentially also replace X with the given definition in terms of T. So we would have 3 Cs T cubed minus i because this is how X is defined. So, going back to our calculation, we have 36 multiplied by 0 squad, multiplied by sin of 3 multiplied by 0 cubed minus pi. So, if we decide to express our derivative in terms of T only, then we don't need to perform any additional calculations for the value of X at the given value of T, right? And from here we can evaluate the result. So we can already see that we have a multiplication by 0. It's simply equal to 0, -36 multiplied by 0, multiplied by sin of 3 multiplied by negative pi. So sign of -3 pi is also 0. So overall we are going to get 0, which corresponds to the answer choice C. Thank you for watching.

Find the value of dy/dt at t = 0 if y = 3 sin 2x and x = t² + π.

Key Concepts

Chain Rule

Implicit Differentiation

Trigonometric Functions

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