Question

Find the derivatives of each of the following functions:

a. 𝑦 = 5𝑥²+7𝑥−8

b. 𝑓(𝑥) = 7/𝑥⁴

c. 𝑦 = 15/√𝑥

d. 𝑓(𝑥) = 2𝑥^7/2−𝑥^-1/3

e. 𝑦 = (4𝑥²−7𝑥)/𝑥

f. 𝑓(𝑥) = 7𝑥³/√𝑥

Answer 1

a.

$$y'=(5𝑥^2+7𝑥−8)'=10x+7$$

b.

$$y'=(\dfrac{7}{x^4})'=\dfrac{7(-4)}{x^5}=-\dfrac{28}{x^5}$$

c.

$$y'=(\dfrac{15}{\sqrt{x}})'=\dfrac{15(-1/2)}{x^{3/2}}=-\dfrac{15}{2x^{3/2}}$$

d.

$$f'(x)=(2𝑥^{7/2}−𝑥^{-1/3})'=2(\dfrac{7}{2})x^{5/2}-(-\dfrac{1}{3})x^{-4/3}$$

$$=7x^{5/2}+\dfrac{1}{3}x^{-4/3}$$

e.

$$y'=((4x^2-7x)/x)'=(4x-7)'=4$$

f.

$$f'(x)=( 7𝑥^3/√𝑥)'=7(x^{5/2})'=\dfrac{35}{2}x^{3/2}$$