Question

A. Use implicit differentiation to find the derivative dy/dx

. b. Find the slope of the curve at the given point. cos (y)=3 x⁴-3

Answer 1

To find dy/dx using implicit differentiation for cos(y) = 3x⁴ - 3, we differentiate both sides and solve for dy/dx, resulting in dy/dx = -12x³ / sin(y). For the slope at a specific point, substitute the point's x and y values into this expression.

Step-by-step explanation:

To use implicit differentiation to find the derivative of dy/dx, given the equation cos(y) = 3x⁴ - 3, we differentiate both sides with respect to x.

Step 1: Differentiate both sides

Take the derivative of cos(y):

d/dx [cos(y)] = -sin(y) (dy/dx)

Now take the derivative of 3x⁴ - 3:

d/dx [3x⁴ - 3] = 12x³

Step 2: Equate and solve for dy/dx

Now we set

-sin(y) (dy/dx) = 12x³,

and solve for dy/dx:

dy/dx = -12x³ / sin(y).

To find the slope of the curve at a specific point, we need to substitute x and y values of that point into the expression for dy/dx.

Example for a given point:

For instance, if the point is (1, π/2), we substitute these values into the dy/dx equation:

dy/dx = -12(1)³ / sin(π/2) = -12