Section 3

Derivative Notation

y=f(x)

Prime Notation

y’ and f’(x)

Leibniz Notation

dy/dx

Power Rule and Constant Rule

Bring exponent to the front(multiply by constant if needed), subtract exponent by 1 to get new exponent to the new constant, change answer into positive exponents if needed.

What does ‘ indicate?

1st Derivative

What does “ indicate?

2nd derivative

If S(t)=Position Function in feet, what is the velocity function?

The derivative of the position function; s’(t) in ft/per unit of time

If S(t)=Position Function in feet, and S’(t) is the velocity function in fat/per unit of time, what is the acceleration function?

The derivative of the velocity function;S”(t) ft/per unit of time^2

How can you determine the points at which a fcn has a horizontal tangent line?

1. Find f’(x)
2. f’(x)=0 (solve)
3. Solve for (x,y) (plug x value found back into the o.g. fcn)

Normal line

The line that is perpendicular to the tangent line at p.

What does perpendicular mean in terms of normal line?

Opposite sign and the reciprocal of the slope of the tangent line

Left hand method

How to remember what trig is positive in what quadrant?

What are the reciprocal identities?

What are the ratio identities?

What are the Pythagorean identities?

What are the special values of quadrant one?

Clockwise on the unit circle gives you what types of pie values?

Negative

Counterclockwise on the unit circle give you what pie values?

Positive

What are the basic derivative rules?

1. Constant function
2. Power rule
3. Constant Multiple
4. Sum rule
5. Difference rule
6. Natural Exponential Fcn

Derivative of a Constant fcn

The Power Rule

The Constant Multiple Rule

The Sum Rule

The Difference Rule

Derivative of the Natural Exponential fcn

The Product Rule

The Quotient Rule

True or False: Order real important in the quotient rule

True

What are two useful trig limits?

What are the steps of the chain rule?

1. Identify the inside function(rewrite o.g. fcn if needed)
2. Identify the outside function
3. Find the d/dx of both
4. Put it together using the general formula
5. Substitute the o.g. inside fcn back in for u
6)Simplify where possible
7)Make sure your answer is has all positive exponents, if not rewrite with positive

Outside fcn with u replacing the existing variables

inside fcn

Derivative of the chain rule

General Power chain rule

General exponential chain rule

General Exponential derivative rule

Root

Power

When you multiply power with the same base you do what with the exponents?

Add them

When you divided powers with the same bases, you do what with the exponents?

Subtract them

Product rule with 3 factors

d/dx \[fgh\]=f’gh+fg’h+fgh’

If you see a trig fcn that has a constant in front of the variable such as sin2t the pattern is:

If you see a trig fcn with a number exponents after the fcn and before the variable such as sin^2x the pattern is:

Other equivalent form of dy/dx

y’x or f’(x)

Finding dy/dx by implicit differentiation

How to find the slope after find dy/dx

Substitute the x value and the y value into the found dy/dx and solve

Domain restriction of sin^-1(x)

Domain restriction of cos^-1(x)

Domain restriction of tan^-1(x)

When should you use the log properties to solve?

Use this when:


1. Variable as base and exponent
2. Very complicated function that you could expand
using logs to make it easier to differenate

What are the steps of logarithmic differentiation?

Pythagorean Theorem

x^2+y^2=z^2 or a^2+b^2=c^2

Volume of a cone

Area of a circle

Trig area formula:

What are the 6 steps to solving related rates problems?