Question

Zenzizenzicube is an obsolete word that represents the square of the square of a cube. In​ symbols, zenzizenzicube is written as

(((x^3)^2))^2

Required:
a. Use the chain rule twice to find the derivative.
b. Use the properties of exponents to first simplify the​ expression, and then find the derivative.

Answer 1

Explanation:

Given the expression (((x^3)^2))^2

a) Differentiating the function with respect to x using chain rule.

Given y = (((x^3)^2))^2

Let u = (x³)²

y = u²; dy/du = 2u

Let p = x³; dp/dx = 3x²

u = p²; du/dp = 2p

Applying the chain rule formula:

dy/dx = dy/du • du/dp • dp/dx

substitute in the formula

dy/dx = 2u•2p•3x²

Since u = (x³)² and p = x³

dy/dx = 2(x³)²(2x³)(3x²)

dy/dx = 12x^6•x^5

dy/dx = 12x¹¹

b) Using property of exponent

y = (((x^3)^2))^2

y = (x^6)²

y = x¹²

dy/dx = 12x^12-1

dy/dx = 12x^11