How to Find the Slope of the Tangent Line Using Implicit Differentiation

Number 45. Find a slope of the tangent line at the point 2a 3. To find the slope of the tangent line, we need to find the value of dydx at the point 2a 3. So let's find dydx. Let's use implicit differentiation. Let's differentiate both sides of the equation with respect to x. The derivative of x cub is 3x^2. Now the derivative of 5x y we need to use the product rule. f * g prime or the derivative of f * g. That's f prime g plus fg prime. So let's say f is 5x, g is y. The derivative of 5x is 5 and then we'll multiply it times g which is y and then minus we need to distribute that negative sign. We have f which is 5x and then the derivative of y with respect to x is 1 * dy dx. Now the derivative of y^2 is 2 y * dy dx and the derivative of the constant 11 is 0. Now for this problem we could solve for dy dx and then plug in the 2a 3 or we could plug it in right now and we could still get the right answer for this problem. Let's go ahead and plug it in right now. 4 * 3 is 12. And then x^2 x is 2 so we'll have 2^ 2 and then - 5 y is 3 and then - 5x x is 2 * dydx and then y is 3. So 2 y or 2 * 3 * dy dx is equal to zero. 2 ^2 is 4. 4 * 12 is 48. 5 * 3 is 15. 5 * 2 is 10. dydx, we can write that as y prime. 2 * 3 is 6. And then there's another y prime next to that. Now, what we can do is combine like terms. 48 - 15, that's going to be 33. -10 y prime + 6 y prime is -4 y prime. Now, I'm going to move this to the other side. It's -4 on the left. It will be pos4 on the right. So we have 33 is equal to 4 y prime. And then I'm going to divide both sides by 4. So y prime which is the same as dydx or dy over dx that's equal to 33 over 4. Now 33 over 4 is not zero and it's not equal to -7 / 6. We need to convert this to a mixed number. What is the highest multiple of four just under 33? That will be 32. So, I'm going to break up 33 into 32 + 1. Both of these will still be divided by 4. So, 33 over 4 is equal to 32 over 4 + 1 over 4. 32 over 4 is 8. 8 + 1 over 4 is 8 and 1/4. So, that's going to be the answer as a mixed number. Therefore, answer choice D is the right answer.