Question

consider 16x² + 49y2 = 784 (a) Find dy / dx by implicit differentiation. dy / dx = 16x 49y (b) Solve the equation explicitly for y and differentiate to get dy / dx in terms of x. (Consider only the first and second quadrants for this par dy / dx =

Answer 1

To find dy/dx, differentiate both sides of the equation and solve for dy/dx. To solve the equation explicitly for y, rearrange and solve for y. Differentiate the equation in terms of x to find dy/dx.

Step-by-step explanation:

To find dy/dx by implicit differentiation, differentiate both sides of the equation with respect to x.

16x^2 + 49y^2 = 784

Take the derivative of each term:

32x + 98yy' = 0

Now solve for dy/dx:

dy/dx = -32x / 98y

To solve the equation explicitly for y, rearrange:

49y^2 = 784 - 16x^2

y^2 = (784 - 16x^2) / 49

y = ±√((784 - 16x^2) / 49)

Now differentiate to find dy/dx in terms of x:

dy/dx = ±(1/49) * (1/2) * (-32x) * ((784 - 16x^2)^(-1/2))

Therefore, dy/dx = ±(16x) / sqrt(784 - 16x^2).