Answer to Question #334163 in Calculus for kristine

Question

find the derivative using chain rule.

1. y= (x³ + 3)⁵

2. y=(-3x⁵ + 1)³

3. y=(-5x³ - 3)³

4. y=(5x² + 3)⁴

5. f(x)=⁴√-3x⁴-2

6. f(x)= √-2x² + 1

Answer 1

$1. y=(x^3+3)^5\\ y'=5(x^3+3)^4(x^3+3)' = 15x^2(x^3+3)^4$

$2. y =(-3x^5+1)^3\\ y'=3(-3x^5+1)^2(-3x^5+1)' = -45x^4(-3x^5+1)^2$

$3. y =(-5x^3-3)^3\\ y'=3(-5x^3-3)^2(-5x^3-3)'= -45x^2(-5x^3-3)^2$

$4. y=(5x^2+3)^4\\ y'=4(5x^2+3)^3(5x^2+3)' = 40x(5x^2+3)^3$

$5. f(x) = \sqrt[4]{-3x^4-2}\\ f'(x)=\cfrac{(-3x^4-2)'}{4\sqrt[4]{(-3x^4-2)^3}}=\cfrac{-3x^3}{\sqrt[4]{(-3x^4-2)^3}}$

$6. f(x) = \sqrt{-2x^2+1}\\ f'(x) = \cfrac{(-2x^2+1)'}{2\sqrt{-2x^2+1}} = \cfrac{-2x}{\sqrt{-2x^2+1}}$