evaluate at x = .5
y(0.5) = 2(.5) + sin(0.5) don't forget to use radians (MODE)
= 1 + .47943
= 1.47943
y'(0.5) = 2 + cos(0.5)
= 2 + .87758
= 2.87758
nDeriv(2x + sin(x),x,0.5) = 2.87758 or
nDeriv(Y1,x,0.5)
nDeriv(Y1,x,x) to plot
For example nDeriv(sin(x),x,x) should plot the cosine

The dy/dx function on the CALC menu just shows the slope of the current graph at the
cursor location. You can also plot the derivative using
Y₁=nDeriv(sin(x),x,x)
The SOLVE function (0 on the MATH menu) looks like:
SOLVE(sin(x)-e^(-x),x,1.,{0,3})
where 1. is the initial guess and 0,3 is the range
The result is x=.58853 sin .58853 = .55514 e^(-.58853) = .55514

The quotient Rule

d(u/v) /dx = (vdu/dx - udv/dx)/ v²

For example:

f(x) = y = cos(x)/sin(x) or y = cot (x)

Let u = cos(x) v = sin(x)

Then calculate

u' = -sin(x) v' = cos(x)

then substitute

y' = (vdu/dx - udv/dx)/ v²

= (sin(x)(-sin(x)) - cos(x)(cos(x)))/ sin² (x)

= -(sin²(x) + cos² (x))/ sin² (x)

= -1/ sin² (x)

y'(.5) = -4.35069

Check the result with nDeriv(tan^-1(x),x,.5)

nDeriv = -4.35070

if you can't remember what gets substracted from what.

try something easy like

d(x/1)/dx

Let u = x and v = 1 then
du/dx = 1 and dv/dx = 0

d(x/1)/dx = ((1)(1) - x (0))/ (1)^{2
= 1}

This is true, the derivative of x is 1

The Chain Rule

dy/dx = (dy/du)(du/dx)
For example
y = sin(4x² + 3x)
Let
u = 4x² + 3x
du/dx = 8x + 3

y = sin(u)

dy/du = cos(u)

dy/dx = (dy/du)(du/dx)

= (cos(u)) (8x+3)

= (cos(4x² + 3x))(8x+3)
Check with x = 0.35 (any number)
y' (.35) = (cos(4x² + 3x))(8x+3)
= (cos ( 1.5400))(5.800)
= (.03079)(5.80)
=.17859
nDeriv(sin(4x² + 3x),x,.35) = .17857

Function Integral
fnInt(expression,variable,lower,upper,tolerance) or
fnInt(expression,variable,lower,upper)

For Example:
Integrate
y = 2x + 1 between 2 and 4
₂⁴ò (2x + 1)dx
 
the integral is x² + x + c
between 4 and 2 is
= (4² + 4 + c) - ( 2² + 2 + c)
= 20 - 6
= 14
fnInt(2x +1,x,2,4) = 14.000

Integrals by guessing and substitution
₂⁴ò sin³3x cos 3x dx

let u = sin 3x
then du=3 cos 3x dx
then substitute

₂⁴ò (1/3)u³du

= u⁴/12
= (1/12)sin⁴(3x) between 2 and 4 = 0.00640

Trig Identities

sin²(x) + cos²(x) = 1
1/sin(x) = csc(x)
1/cos(x) = sec(x)
1/tan(x) = tan(x)^-1 = cot (x) does not equal tan^-1(x)
cos²(x) = (1 + cos(2x))/2
sin²(x) = (1 - cos(2x))/2

sin(A + B) = sinA cos B + cos A sin B
if A = B
you end up with
sin 2A = sin A cos A + cos A sin A
 = sin A cos A + sin A cos A (cumunitive property of multication)
 = 2 sin A cos A ( the double-angle formula)

cos(A + B) = cos A cos B - sin A sin B
if A = B
cos 2A = cos A cos (A) - sin A sin (A)
= cos²A - sin²A ( the double-angle formula)

tan (A + B) = sin (A + B)/cos(A + B)
= (tan A + tan B)/(1 - tan A tan B)

Laws of Logarithms
If M and N are positive real numbers and b is a positive number other than 1, then:

Laws of Exponents