Question

Scoring r last submission is used for your score. [-/1 Points] Find dy/dx by implicit differentiation. (sin πx+cos πy)^6=31 dy/dx=

Answer 1

To find dy/dx by implicit differentiation for the equation (sin πx + cos πy)^6 = 31, apply the chain rule, differentiate the inner and outer functions, then solve for dy/dx to arrive at the formula.

Step-by-step explanation:

To find dy/dx by implicit differentiation for the equation (sin πx + cos πy)^6 = 31, we differentiate both sides with respect to x.

First, applying the chain rule:

1. Differentiate the outer function, which is the 6th power.
2. Multiply it by the derivative of the inner function sin πx + cos πy with respect to x.

Here is how we differentiate step by step:

1. The derivative of the left side with respect to x is 6(sin πx + cos πy)^5 * (πcos πx - πsin πy*dy/dx).
2. The derivative of the right side with respect to x is 0 as 31 is a constant.
3. After differentiating, we solve for dy/dx.

So the equation after differentiation is 6(sin πx + cos πy)^5 * (πcos πx - πsin πy*dy/dx) = 0. Solving for dy/dx, we get:

dy/dx = (πcos πx*6(sin πx + cos πy)^5)/(6πsin πy*(sin πx + cos πy)^5).