13-26 Implicit differentiation Carry out the following steps.

Question

b. Find the slope of the curve at the given point.

(x+y)^2/3=y; (4, 4)

Accepted Answer

Welcome back, everyone. Consider the equation of the curve given by X minus Y raise to the power of 3/4 equals X. Calculate the slope of the line, tension to the curve at the 0.10. We're given 4 answer choices A says -1/3, B-3, C 34, and D-4/3. So what we're going to do is simply evaluate the derivative of this function and calculate the slope at the given point, right? So, we're given X minus Y. Raise to the power of 3/4 equals x and we're going to perform implicit differentiation. So let's take the derivative of the left hand side. First of all, the outer function has an exponent, so we bring down the exponent 34s. we multiply that by x minus y, erase the power of 3/4 minus 1 based on the power rule. This gives us 1/4, and then based on the chain rule, we have to multiply that by the derivative of X minus Y. And this must be equal to the derivative of the right hand side, so essentially the derivative of X is 1. Now let's simplify. So we're going to get 3/4s multiplied by x minus Y, raise the power of -1/4, which is then multiplied by the derivative of X, which is 1 minus the derivative of Y. And this is equal to 1, right? So now what we can do is simply solve for Y because this is the derivative. Before we do that, let's simply rewrite the function as we. Divided by Now let's understand that we have 4 in the denominator and now also we have a negative exponent. So this means that we can write X minus Y in the denominator with a positive exponent, raise the power of 14, right? And now in the numerator we have 3 multiplied by 1 minus the derivative of Y. And this is equal to one. Now, what we're going to do from here is simply solve for Y. We can multiply both sides by the denominator, making sure that it is not equal to 0, right? So, essentially, we are going to write that 3 multiplied by 1 minus Y equals 4. Multiplied by x minus Y was the power of 1/4 such that x is not equal toy because this will give us a denominator of 0. And now we understand that the denominator is never going to be zero in the domain of the given derivative, right, because X minus Y is in the denominator based on differentiation. So now from here, let's distribute the terms. 3 minus 3 Y is equal to 4 multiplied by X minus Y, raise the power of 1/4. Now what we want to do is simply We write it as 3 minus the right hand side term, so that's 3 minus or multiplied by X minus Y raise the power of 1/4 is equal to. Now let's move 3y to the right hand side. So instead of being negative, it Is going to become positive and now let's divide both sides by 3, so that Y is equal to 3 minus 4. Multiplied by x minus Y raises the power of 14th divided by 3. And now we want to evaluate the slope of the tangent line, so we want to evaluate at. The 10, so X becomes 1, or Y becomes 0. We're going to get 3 minus 4 multiplied by X minus Y. So 1 minus 0. Raise the power of 14th divided by 3. Now, 1 minus 0 gives us 11 raises the power of 14th is just 1. So 3 minus 4 multiplied by 1. Gives us -1, and then we have 3 in the denominator. So the answer to this problem is -1/3, which corresponds to the answer choice A. Thank you for watching.

13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
(x+y)^2/3=y; (4, 4)

Key Concepts

Implicit Differentiation

Slope of a Curve

Chain Rule

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