(cos πx + sin πy)4 = 54; Implicit Differentation
clarification needed:
is it
(cos πx + sin πy)^4 = 54
or
(cos πx + sin πy)(4) = 54 ?
(cos πx + sin πy)4 = 54
well, when you finally figure it out, Kris, just remember that
-π cosπx + π cosπy y'
is the derivative of what's in the parentheses.
To find the derivative of an equation using implicit differentiation, we'll differentiate both sides of the equation with respect to the variable(s) involved. Let's start by differentiating each term separately.
Given equation: (cos πx + sin πy)^4 = 54.
1. Differentiate the left-hand side:
To differentiate (cos πx + sin πy)^4, we'll use the chain rule. Let's denote this term as u.
Let u = (cos πx + sin πy)^4.
Now, we'll differentiate both sides of the equation with respect to x.
du/dx = d/dx[(cos πx + sin πy)^4].
Applying the chain rule, we have:
du/dx = 4(cos πx + sin πy)^3 × d/dx(cos πx + sin πy).
2. Differentiate the right-hand side:
The right-hand side of the equation, which is 54, is a constant. The derivative of any constant with respect to any variable is always zero.
Now, let's differentiate d/dx(cos πx + sin πy), which is the derivative of the term inside the parentheses.
To do this, we'll differentiate cos πx separately with respect to x and sin πy separately with respect to y. The variables x and y are treated as independent variables.
d/dx(cos πx) = -π sin πx, using the chain rule.
d/dy(sin πy) = π cos πy, using the chain rule.
3. Putting it all together:
du/dx = 4(cos πx + sin πy)^3 × (-π sin πx) + 4(cos πx + sin πy)^3 × π cos πy.
Simplifying this expression, we have:
du/dx = -4π(sin πx)(cos πx + sin πy)^3 + 4π(cos πy)(cos πx + sin πy)^3.
The derivative of the left-hand side is du/dx, and the derivative of the right-hand side is 0, as it's a constant.
Therefore, the final derivative is:
-4π(sin πx)(cos πx + sin πy)^3 + 4π(cos πy)(cos πx + sin πy)^3 = 0.
This is the result obtained by applying implicit differentiation to the given equation.