Find the derivative of the function by using the Product Rul

Find the derivative of the function by using the Product Rule. Simplify your answer. f(z) = (16z4 + 4z2 + 1)(4z3 z)

Solution

given f(z) = (16z⁴ + 4z² + 1)(4z³ z)

differentiate with respect to z on both sides. product rule:(uv)\'=u\'v +uv\' ,d(xⁿ)_/dx =nx^n-1

f \'(z) = [(16*4z^4-1 + 4*2z^2-1 + 0)(4z³ z)]+[(16z⁴ + 4z² + 1)(4*3z^3-1 1)]

f \'(z) = [(64z³ + 8z)(4z³ z)]+[(16z⁴ + 4z² + 1)(12z² 1)]

f \'(z) = [256z⁶ + 32z⁴-64z⁴ 8z²]+[192z⁶ + 48z⁴ + 12z²-16z⁴ - 4z² - 1]

f \'(z) = 448z⁶   1