13-26 Implicit differentiation ...

Question

13-26 Implicit differentiation Carry out the following steps.

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

x⁴+y⁴ = 2;(1,−1)

Accepted Answer

Welcome back, everyone. Given the equation X cubed plus Y cubed equals 9, use implicit differentiation to find the slope of the tangent line at the point 2.1. We're given 4 answer choices A says -14, B 1/4, C-4, and D4. So to find the slope of the tangent line, well, essentially we want to identify the first derivative of the function, and we're going to use implicit differentiation. This essentially means that we're going to rewrite the given equation, X cubed plus Y cubed equals 9, and we're going to take the first derivative of both sides. So now if we distribute the derivative on the left hand side, we are differentiating X cubed. We are adding the derivative of Y cubed, and we're setting it equal to the derivative of 9. Now let's identify all three derivatives starting with X cubed. The derivative is 3x2d based on the power rule because we bring down the exponent and decrease the original exponent by 1 unit. Now for Y cubed we apply the same rule which gives us 3 Y squad, but Y is a function of X, so we apply the chain rule and we multiply by the derivative of Y, which is Y. And now the derivative of 9 is 0 because it is the derivative of a constant. And now if we substitute the results into our expression, we're going to get 3 x 2 plus 3 Y 2 Y equals 0. And now let's so 4 Y. If we move 3x2 to the right, it is going to become negative 3X2. And then we are dividing by 3 Y quad 24 Y. We can notice that we can cancel out the 3. And then our y becomes negative X2 divided by Y2. And this is the equation of a slope of a tangent line at any point x y. And our X in the given problem is 2. Our Y is equal to 1. This is the given point of tangency, meaning the slope at 2.1 is equal to x2, so that's 22 divided by. Y squared, so that's a 1 squared. And when we simplify, we get 4 divided by 1, which is 4, and then the negative sign gives us -4. This corresponds to the answer of choice C. Thank you for watching.

13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.x⁴+y⁴ = 2;(1,−1)

Key Concepts

Implicit Differentiation

Slope of a Curve

Chain Rule

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