Question

Josiah is working with a friend to find the the implicit derivative of the function xy³ + eˣ =sin(y) Josiah finds the answer

Answer 1

The implicit derivative of the function xy³ + eˣ =sin(y) is given by y’ = (-3xy² - eˣcos(y))/(x + cos(y)).

Step-by-step explanation:

To find the implicit derivative of the given function, we start by differentiating both sides of the equation with respect to x. The derivative of xy³ with respect to x is y³ + x(3y²y’), using the product rule. The derivative of eˣ with respect to x is simply eˣ. The derivative of sin(y) with respect to x is cos(y)y’.

After differentiating, we rearrange the terms to solve for y’. This involves isolating y’ on one side of the equation. We then factor out y’ and solve for it to obtain the implicit derivative.

Substituting the derivatives and rearranging, we get y’ = (-3xy² - eˣcos(y))/(x + cos(y)), which represents the implicit derivative of the given function with respect to x.