What is the derivative of y = x - Cos(x)? and why?

Question

specific steps please

Answer 1

dy/dx = 1 + sin(x)

x goes to 1 because you drop the power by 1
x^n goes to nx^(n-1)

cos goes to -sin
sin goes to cos

Answer 2

dy/dx= 1 + sin(x)

you find the derivative of each term individually. the derivative of x is 1 from the formula dy/dx= n*x^(n-1). the derivative of cos(x) is -sin(x), and that's just something you have to memorize. that's all there is to it.

Answer 3

Let us derive cos(x)

The derivative is

lim(h -> 0) [cos(x + h) - cos(x)]/h =
= lim (h -> 0)[cos (x) cos (h) - sin(x) sin(h) - cos(x)]/h =
= lim (h -> 0){[cos (x)(cos(h) - 1] - sin(x)sin(h)]}/h = (*)

Now, as h approaches zero cos(h) tends to 1 and sin(h)/h tends to 1:

(*) = lim(h -> 0) (cos(x)(1 - 1) sin(x)sin(h))/h =
= lim(h->0) -sin(x)sin(h)/h = -sin(x)

From the linearity of limits - particularly derivatives - we can
subtract this derivative from 1, the derivative of x:
y' = 1 + sin(x)

Answer 4

d(y)/dx or y' will be written as 1+sinx.

1st derivative of 'x' will be 1; cos(x) is -sin(x). So your equation is as follows:

y=x-cos(x)
y'=1+sin(x).

Answer 5

use implicit differentiation so dy/dx=1+sin(x)