Question

Let y be defined implicitly by the equation 4x³ −5y⁴−5x−7y=1 Use implicit differentiation to evaluate dx/dy at the point (−1,0).

Answer 1

Using implicit differentiation on the equation 4x³ −5y⁴−5x−7y=1 yields that dx/dy equals 1 at the point (−1,0).

Step-by-step explanation:

Let y be defined implicitly by the equation 4x³ −5y⁴−5x−7y=1. To evaluate dx/dy at the point (−1,0) using implicit differentiation, we differentiate both sides of the equation with respect to y, remembering to use the chain rule for every term involving x.

First, we differentiate each term:

• For 4x³, we get 12x² * dx/dy (chain rule).
• For −5y⁴, the derivative is −20y³.
• For −5x, the derivative is −5 * dx/dy (chain rule).
• For −7y, the derivative is −7.

Now, putting it all together:

12x² * dx/dy - 20y³ - 5 * dx/dy - 7 = 0

Combine like terms and factor out dx/dy:

(12x² - 5) * dx/dy = 20y³ + 7

To isolate dx/dy:

dx/dy = (20y³ + 7) / (12x² - 5)

Substituting x = −1 and y = 0 gives:

dx/dy = (20(0)³ + 7) / (12(−1)² - 5)

Simplify the right side:

dx/dy = 7 / (12 - 5)

dx/dy = 7 / 7

dx/dy = 1

Thus, the value of dx/dy at the point (−1,0) is 1.