Question

This is for AP CALCULUS. I really need help on this. I’m trying to find the first and second derivative of these three functions.

Answer 1

Review Material:

`\sqrt[n]{x^m} = x^{(m)/(n)}`

`(d)/(dx)[x^n]=nx^(n-1)\\`

`(d)/(dx)[\sin(x)]=\cos(x)\\`

`(d)/(dx)[\cos(x)]=-\sin(x)\\`

`(d)/(dx)[constant]=0\\`

Explanation:

(a)

`F(x) = -2\cos(x) +x^{(4)/(3)} -3e\\ \text{Note: 3e is a constant}\\F'(x) = 2\sin(x) + (4)/(3)x^{(1)/(3)}\\F''(x) = 2\cos(x) + (4)/(9)x^{-(2)/(3)}`

(b)

`F(x) = x^(-3) + (1)/(2)x^2-\sin(x)\\F'(x)=-3x^(-4) + x - \cos(x)\\F''(x) = 12x^(-5) + 1 + \sin(x)`

(c)

`F(x) = 2x^(-3)+x^{(3)/(4)}-4x\\F'(x) = -6x^(-4) + (3)/(4)x^{-(1)/(4)}-4\\F''(x)= 24x^(-5) - (3)/(16)x^{-(5)/(4)}`