Calculus Derivatives Study Guide

Limit Definition
d/dx = lim(∆x→0) f(∆x+x) – f(x) / ∆x
Finding points where tangent line to function is horizontal. (Horizontal tangent line to a function)
1. Take derivative
2. Derivative=0
3. Solve for x
4. Plug x into original for y
Equation of line normal to function at given point.
1. Take derivative
2. Plug given x value into derivative (for slope)
3. Negative reciprocal of slope
4. Plug into point-slope form
Evaluating Trig Functions: Sin
sin0 = 0
sin(π/6) = 1/2
sin(π/4) = √2/2
sin(π/3) = √3/2
sin(π/2) = 1
Evaluating Trig Functions: Cos
cos0 = 1
cos(π/6) = √3/2
cos(π/4) = √2/2
cos(π/3) = 1/2
cos(π/2) = 0
Evaluating Trig Functions: Tan
tan0 = 0
tan(π/6) = √3/3
tan(π/4) = 1
tan(π/3) = √3
tan(π/2) = U
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Evaluating Trig Functions: Csc
csc0 = U
csc(π/6) = 2
csc(π/4) = √2
csc(π/3) = 2/√3
csc(π/2) = 1
Evaluating Trig Functions: Sec
sec0 = 1
sec(π/6) = 2/√3
sec(π/4) = √2
sec(π/3) = 2
sec(π/2) = U
Evaluating Trig Functions: Cot
cot0 = U
cot(π/6) = √3
cot(π/4) = 1
cot(π/3) = √3/3
cot(π/2) = 0
Find value of k so function f(x) is tangent to a line.
1. Find derivatives of both equations
2. Set derivatives = (slopes are =)

3. Set originals = (graphs intersect)

4. Decide which equation is easier to solve for either k
5. Substitute k value into other equation (only x left)
6. Solve for x

7. Plug x value into equation to find k

Functions aren’t differential if…
-Vertical tangent lines
-Sharp turns
Derivative of: sin(x)
cos(x)
Derivative of: cos(x)
-sin(x)
Derivative of: tan(x)
sec²(x)
Derivative of: sec(x)
sec(x)tan(x)
Derivative of: cot(x)
-csc²(x)
Derivative of: csc(x)
-csc(x)cot(x)
Finding d²y/dx²
1. Solve for dy/dx
2. Take derivative of dy/dx
3. Plug in dy/dx into derivative of dy/dx (possible fraction buster)
Average rate of change/average velocity
f(b)-f(a) / b-a
Instantaneous rate of change/instantaneous velocity
f'(x)
What values does average velocity = instantaneous velocity.
1. Set avg. & inst. equations =
2. Plug in equations as Y1 & Y2 into graphing calculator
3. Set window using given interval
4. Graph → calc → intersections
5. Make sure x values are within interval
Volume of Sphere
V = (4/3)πr³
Surface Area of Sphere
SA = 4πr²
Volume of Right Circular Cylinder (& General Prisms)
V = Bh
Volume of Square Pyramid
V = (1/3)Bh
Volume of Right Circular Cone
V = (1/3)πr²h