Calculus 1 formula sheet
Calculus
Front Range Community College
2 pag.
Document shared on https://www.docsity.com/en/calculus-1-formula-sheet/8254843/
Calculus I
Formula Sheet
Chapter 3
Downloaded by: mk-ndlovu-1 (sihle98sc@outlook.com)
Section 3.3
Section 3.1
1. Definition of the derivative of a function:
f ( x + ∆x) − f ( x)
f ′ ( x ) = lim
∆x→ 0
∆x
2. Alternative form of the derivative at x = c :
f ( x ) − f (c )
f ′ ( c ) = lim
x→ c
x−c
3. Differentiability ⇒ Continuity
Section 3.2
d
4.
[c] = 0
dx
d
5.
[ x] = 1
dx
d  n
6.
x = n x n −1
dx  
d
7.
 c f ( x )  = c f ′ ( x )
dx 
d
8.
[ f ( x) + g ( x)] = f ′( x) + g ′( x)
dx
d
9.
[ f ( x) − g ( x)] = f ′( x) − g ′( x)
dx
d
10.
[sin x ] = cos x
dx
d
11.
[cos x ] = − sin x
dx
d  x
12.
e = ex
dx  
1
13. Free Fall: s(t ) =
− gt 2 + v0t + s0
2
Feet: -32
Meters: -9.8
14. Velocity: v ( t ) = s′(t )
15. Find equation of tangent line to curve at
x = c:
a. Equation of Line: need point and slope
b. Point: ( c, f ( c ) )
Slope: m = f ′(c)
c. Point-Slope form: y − y1= m ( x − x1 )
16. Derivative:
a) Slope of tangent line
b) Instantaneous rate of change
d
x )] f ( x ) g ′ ( x ) + g ( x ) f ′ ( x )
[ f ( x ) ⋅ g (=
dx
d  f ( x)  g ( x ) f ′ ( x ) − f ( x ) g′ ( x )
18.
=
dx  g ( x ) 
[ g ( x )]2
17.
d
[ tan x ] = sec2 x
dx
d
20.
[sec x ] = sec x tan x
dx
d
21.
[cot x ] = − csc2 x
dx
d
22.
[csc x ] = − csc x cot x
dx
′( x) 0
23. Horizontal Tangent Line:=
m f=
19.
Section 3.4
24. Chain Rule:
=
y y (u ), =
u u( x ),
dy dy du
=
⋅
dx du dx
d
1
[ln x ] =
dx
x
26. Definition of exponential function to base a:
25.
a x = e( )
27. Definition of logarithmic function to base a:
1
log a x =
ln x
ln a
d  x
28.
a = ( ln a ) a x
dx  
d
1
29.
[loga x ] =
dx
(ln a ) x
ln a x
Section 3.5
30. Implicit Differentiation:
a. Differentiate both sides with respect to x.
dy
on the left side.
b. Isolate
dx
31. Logarithmic Differentiation
a. Take a ln of both sides
b. Use properties of ln to expand
c. Differentiate implicitly
Document shared on https://www.docsity.com/en/calculus-1-formula-sheet/8254843/
Section 3.6
( )
′
32. Find f −1 (b)
a. Formula:
b.
( f −1 )′ (b) = f ′ f 1−1 (b)
)
(
f −1 ( b ) =
a ⇒ f (a ) =
b
c. Set f ( x ) = b and solve.
d. Find f ′ ( x ) at the solution found in
the previous step.
e. Apply the formula at the beginning.
d
1
33.
[arcsin x ] =
dx
1 − x2
d
1
34.
[arctan x ] =
dx
1 + x2
1
d
35.
[arcsec x ] =
dx
x x2 − 1
−1
d
[arccos x ] =
dx
1 − x2
−1
d
37.
[arc cot x ] =
dx
1 + x2
−1
d
38.
[arc csc x ] =
dx
x x2 − 1
36.
Section 3.7
39. Related Rate Problems
a. Identify all given and to find
b. Equation
c. Differentiate Implicitly
d. Substitute and solve
Document shared on https://www.docsity.com/en/calculus-1-formula-sheet/8254843/