. For each of the following functions, find the indicated derivative. Solve using either the definition of the derivative or its alternate definition.

Question

For each of the following functions, find the indicated derivative. Solve using either the definition of the derivative or its alternate definition.

a. 𝑓(𝑥) = 4𝑥³ − 3𝑥² − 2𝑥 − 1 ; 𝑓 ′ (−1) d. 𝑓(𝑥) = 𝑥 sin 𝑥 ; 𝑓 ′ (𝜋)

b. 𝑓(𝑥) = 𝑥/x + 1 ; 𝑓 ′ (4) e. 𝑓(𝑥) = e^x+1 ; 𝑓 ′ (0)

c. . 𝑓(𝑥) = ³√𝑥 ; 𝑓 ′ (8) f. 𝑓(𝑥) = x /1 + sin(𝜋𝑥) ; 𝑓 ′ ( 1/2 )

Answer 1

f(x) = 4x^3 -3x^2 -2x -1 f'(-1) =?

f'(x) = 12x^2 -6x -2

f'(-1) = 12(-1)^2 -6(-1) = 12+6 = 18

f(x) = x/(x+1) f'(4) =?

f'(x) = [(x+1) + x]/(x+1)^2 = (2x+1)/(x+1)^2

f'(4) = (2(4)+1)/(4+1)^2 = 9/25

f(x)=cube root of x f'(8)= ?

f'(x) = (1/3)x^(-2/3)

f'(8) = (1/3)(8)^-(2/3) = 1/3(8)^2/3 = 1/3(64)^1/3 = 1/3(4) = 1/12

f(x) = xsinx f'(pi) = ?

f'(x) = xcosx + sinx (derivative of a product is 1st term times derivative of the 2nd term + the 2nd

term times derivative of the 1st term)

f'(pi) = pi(cos(pi)) + sin(pi) = pi(-1) + 0 = -pi

f(x) = e^(x+1) f'(0) = ?

f'(x) = e^(x+1)

f'(0) = e^(0+1) = e^1 = e