In Exercises 51 and 52, find dp/dq.

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Question

In Exercises 51 and 52, find dp/dq.

p³ + 4pq - 3q² = 2

Accepted Answer

Welcome back, everyone. Find DM divided by D and by implicit differentiation. We're given M cubed plus 6 minus 2N2d equals 5. Let's begin by understanding that M is a function of N, right, because we're looking for DM divided by DN. So what we're going to do is simply differentiate the left hand side M cubed plus 6. Minus 2 n squad and set this derivative equal to the derivative of the right hand side, so the derivative of 5 and now we can apply the rules of differentiation. First of all, we will need to differentiate m cubed. What is the derivative of M cubed? Well, it is a composite function, right, because it's a function of m and we have the outer function which is the cube, right? So according to the chain rule, first of all, we apply the power rule. So the derivative of M cubed is going to be 3 m2, which is then multiplied by the derivative of M. So here we have applied the chain rule, right? Because first of all, we have differentiated M cube with respect to M, and then we have to multiply by the derivative of M because M is a function of M. Now we are adding 6 multiplied by the derivative of mm. Here we're going to apply the product rule which says that we're taking the derivative of M, multiplying by N and adding M multiplied by the derivative of n. We're going to leave M prime as it is. But the derivative of N is equal to 1 because that's our independent variable, so we get M N plus M. And then we also have to differentiate minus 2 and 2, right? So the derivative of N2d is going to be equal to 2N because that's our independent variable and we are applying a power rule and we're setting it equal to the derivative of 5. So now let's substitute all of the results. We M 2 M + 6 in parent we have our product, so that's mm plus M, and we are subtracting 2 multiplied by 2n, which is 4 N. and this is going to be equal to 0 because the derivative of 5 is 0, right? It's a constant. We're going to distribute our terms we M 2 M. + 6 M N + 6 M minus 4N is going to be equal to 0. Now we want to leave all of the terms involving M on the left, and we're going to move the other two terms to the right because we're going to solve for M. Let's recall that M is simply DM divided by DN. It's just a different notation, but this is exactly what we're solving for. So if we perform factorization, we can factor out M. Which leaves us with 3 m2 + 6M. And this is equal to. Or if we move negative for n to the right, it becomes positive, minus 6M we're also moving a positive term to the right, so it becomes negative. And now dividing the right hand side. By the factor on the left hand side, we end up with M equals 4 N minus 6M divided by. VM 2 + 6M, which is DM divided by DM, and that's our final answer, so let's label it and thank you for watching.

In Exercises 51 and 52, find dp/dq.

p³ + 4pq - 3q² = 2

Key Concepts

Implicit Differentiation

Chain Rule

Partial Derivatives

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