Answer to Question #306133 in Calculus for Anniee

Question

Find the dy/dx and simply the result, if possible

a. y=√x-1/√x

b. y=x²+π²+xπ

c.y=x² sec x

d.y=sinx-1/cosx

e.y=1/ex+2

Answer 1

a.

$$y'=(\sqrt{x}-\dfrac{1}{\sqrt{x}})'=\dfrac{1}{2\sqrt{x}}+\dfrac{1}{2x\sqrt{x}}$$

$$=\dfrac{x+1}{2x\sqrt{x}}=\dfrac{\sqrt{x}(x+1)}{2x^2}$$

b.

$$y'=(x^2+\pi^2+x\pi)'=2x+\pi$$

c.

$$y'=(x^2\sec x)'=2x\sec x+x^2(-\dfrac{1}{\cos ^2 x})(-\sin x)$$

$$=2x\sec x+x^2\sec x\tan x$$

d.

$$y'=(\sin x-\dfrac{1}{\cos x})'=\cos x-(-\dfrac{1}{\cos ^2 x})(-\sin x)$$

$$=\cos x-\dfrac{\tan x}{\cos x}$$

e.

$$y'=\dfrac{1}{e^x+2}=-\dfrac{e^x}{(e^x+2)^2}$$