Find dy/dx by implicit differentiation.
(sin(pix)+cos(pix))^7=29
cos (pi x) / sin (pi y)
To find dy/dx by implicit differentiation, we need to differentiate both sides of the equation with respect to x, treating y as a function of x and using the chain rule whenever necessary.
Let's take the equation (sin(pix) + cos(pix))^7 = 29 and differentiate it term by term.
Differentiating both sides, we get:
d/dx[(sin(pix) + cos(pix))^7] = d/dx[29]
Now, let's focus on differentiating the left side using the chain rule. We can rewrite it as:
[d/dx(sin(pix) + cos(pix))]^7
Applying the chain rule to the inner function sin(pix) + cos(pix), we have:
d/dx(sin(pix) + cos(pix)) = (d/dx(sin(pix)) + d/dx(cos(pix)))
The derivative of sin(pix) with respect to x can be found using the chain rule again:
d/dx(sin(pix)) = (d/d(pix))(d/dx(pix))
Since d/d(pix) is just a constant, we can replace it with a placeholder, let's say k:
= k * (d/dx(pix))
Similarly, we can find the derivative of cos(pix) with respect to x:
d/dx(cos(pix)) = (-sin(pix))(d/dx(pix))
= -sin(pix) * (d/dx(pix))
Substituting these expressions back into our differentiation of the left side, we have:
[d/dx(sin(pix) + cos(pix))]^7 = (k * (d/dx(pix)) - sin(pix) * (d/dx(pix)))^7
= [k * (d/dx(pix)) - sin(pix) * (d/dx(pix))]^7
Now, let's simplify the right side, which is 29. The derivative of a constant with respect to x is 0, so the entire right side becomes 0.
Therefore, we have:
[k * (d/dx(pix)) - sin(pix) * (d/dx(pix))]^7 = 0
To find dy/dx, we want to solve for d/dx(pix). So, rearranging the equation:
[k * (d/dx(pix)) - sin(pix) * (d/dx(pix))]^7 = 0
Taking the seventh root of both sides to isolate the term inside the brackets:
k * (d/dx(pix)) - sin(pix) * (d/dx(pix)) = 0
Factoring out the derivative term:
(d/dx(pix))(k - sin(pix)) = 0
From this equation, we see that either d/dx(pix) = 0 or k - sin(pix) = 0. Let's solve for each case individually:
Case 1: d/dx(pix) = 0
This case implies that x is independent of y, so dy/dx = 0.
Case 2: k - sin(pix) = 0
Solving for k:
k = sin(pix)
Substituting this back into our earlier expression:
d/dx(pix) = sin(pix)
Therefore, d/dx(pix) = sin(pix) is the other possible value.
In summary, there are two possible values for dy/dx:
1. dy/dx = 0 (when d/dx(pix) = 0)
2. dy/dx = sin(pix) (when k - sin(pix) = 0)