Question

Find d y / d x by implicit differentiation. r sin (x)+cos (y)=9 x-8 y d y / dx =9-cos (x)-sin (y)+8]

Answer 1

To find dy/dx by implicit differentiation, differentiate both sides of the equation with respect to x by using the chain rule. Isolate d(y)/dx to find the final answer.

Step-by-step explanation:

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

Let's start by differentiating the left side using the chain rule:

d(rsin(x) + cos(y))/dx

= d(rsin(x))/dx + d(cos(y))/dx

Using the chain rule, d(rsin(x))/dx = cos(x) * d(rsix(x))/dx and d(cos(y))/dx = -sin(y) * d(y)/dx.

So, the left side becomes: cos(x) * d(rsix(x))/dx - sin(y) * d(y)/dx.

Now, let's differentiate the right side with respect to x. The derivative of 9x - 8y with respect to x is simply 9.

Putting it all together, we have:

cos(x) * d(rsix(x))/dx - sin(y) * d(y)/dx = 9.

Now we can solve for d(y)/dx by isolating it on one side of the equation. The final answer is d(y)/dx = (9 - cos(x) * d(rsin(x))/dx) / sin(y).