Question

Find dy/dx by implicit differentiation.

(cos πx + sin πy)5 = 54

show work please:)

Answer 1

To find dy/dx by implicit differentiation, differentiate both sides of the equation with respect to x using the chain rule. Solve for dy/dx by simplifying the equation.

Step-by-step explanation:

To find dy/dx by implicit differentiation, we will differentiate both sides of the equation with respect to x. Starting with the left side of the equation, we can apply the chain rule to the function (cos πx + sin πy)⁵:

d/dx((cos πx + sin πy)⁵) = d/dx(54)

Using the chain rule, the derivative of (cos πx + sin πy)⁵ with respect to x is 5(cos πx + sin πy)⁴(-πsin πx - πcos πy(dy/dx)). The derivative of 54 with respect to x is 0 because it's a constant. Simplifying this equation, we get:

5(cos πx + sin πy)⁴(-πsin πx - πcos πy(dy/dx)) = 0

From here, we can solve for dy/dx:

dy/dx = -πsin πx / (πcos πy)

Answer 2

This is the concept of calculus, to get dy/dx we proceed as follows;
(cos πx+ sin πy)5=54
first we divide through by 5 we get:
(cos πx+ sin πy)=10.8
differentiating our expression we get:
-sin πx+y'cos πy=0
this can be written as:
y'cos πy=sin πx
dividing both sides by cos πy we get:
y'=[sin πx]/[cos πy]
The answer is y'=[sin πx]/[cos πy]