Find d/dx for (sin πx + cos πy)4 = 65
To find d/dx for the given equation, we need to take the derivative with respect to x on both sides of the equation.
Let's start by differentiating the left-hand side of the equation. We can use the chain rule to differentiate (sin πx + cos πy)^4 with respect to x.
Let's denote the function f(x, y) = (sin πx + cos πy)^4. Applying the chain rule, the derivative of f with respect to x can be written as:
d/dx [(sin πx + cos πy)^4] = 4(sin πx + cos πy)^3 * (d/dx [sin πx + cos πy])
To proceed further, let's differentiate the terms sin πx and cos πy separately with respect to x:
d/dx [sin πx] = π * cos πx
d/dx [cos πy] = 0 (since cos πy is not dependent on x)
Now, substitute these values back into the chain rule expression:
d/dx [(sin πx + cos πy)^4] = 4(sin πx + cos πy)^3 * (π * cos πx + 0)
= 4π(sin πx + cos πy)^3 * cos πx
Now, we have the derivative of the left-hand side of the equation with respect to x. Since the equation states that this derivative is equal to 65, we have:
4π(sin πx + cos πy)^3 * cos πx = 65
And this is the differential equation we have obtained by taking the derivative with respect to x.