Answer:
dz/dx = -4x^3y^7y^8sin(x^4y^8) * (2 + 8x^3y^8dy/dx)
Explanation:
Using the chain rule, we can find dz/dx in terms of x, y, and dy/dx:
dz/dx = dz/dy * dy/dx + dz/dx
Where dz/dy is the partial derivative of z with respect to y, and dz/dx is the partial derivative of z with respect to x.
To find dz/dy, we can use the chain rule again:
dz/dy = -sin(x^4y^8) * d/dy (x^4y^8)
Using the product rule and the chain rule, we get:
d/dy (x^4y^8) = x^4 * d/dy (y^8) + y^8 * d/dy (x^4)
= 8x^4y^7 * dy/dy + 4x^3y^8 * 1
= 8x^4y^7 + 4x^3y^8
Substituting this into the expression for dz/dy, we get:
dz/dy = -sin(x^4y^8) * (8x^4y^7 + 4x^3y^8)
= -4x^3y^8sin(x^4y^8) * (2y + x * y^7)
To find dz/dx, we can now substitute the expressions we found for dz/dy and d/dx (cos (x^4y^8)):
dz/dx = -sin(x^4y^8) * d/dx (x^4y^8)
= -sin(x^4y^8) * (4x^3y^8 * dy/dx + x^4 * 8y^7 * dy/dx)
= -4x^3y^7y^8sin(x^4y^8) * (2dy/dx + x * 8y^7dy/dx)
= -4x^3y^7y^8sin(x^4y^8) * (2 + 8x^3y^8dy/dx)
Therefore, dz/dx in terms of x, y, and dy/dx is:
dz/dx = -4x^3y^7y^8sin(x^4y^8) * (2 + 8x^3y^8dy/dx)