Derivative ToolKit

Question 1

What must be true for a limit to exist?

Answer 1

The limit from the left of the variable must be equal to the limit from the right of the variable.

Question 2

What is a limit at point a generally equal to?

Answer 2

The function value at point a.

Question 3

What methods can be used if a limits answer is 0/0?

Answer 3

-Factorise the limit to get a different answer.
-Multiply the limit by the conjugate over itself.

Question 4

What method can be used if a limit’s answer is 0/0 and there is an absolute value present?

Answer 4

-Through use of the definition of the abs val.
-If approaching from the left x will be less than what it is approaching and therefore the abs val will be -(x).
-The opposite is also true.
-Through this method the function can be factorised.

Question 5

What must be done is a limit gives the answer a/0?

Answer 5

-Approaching from the right anything divided by 0 is infinity.
-Approaching from the left anything divided by 0 is negative infinity.

Question 6

What method must be used if 0 x infinity occurs in a limit?

Answer 6

Factorise that which becomes 0 and then multiply out before substituting x in.

Question 7

What must be done if a limit at infinity is used on a polynomial?

Answer 7

Substitute out the highest degree of x and then substitute in infinity.

Question 8

What must be done when a square root is present in a limit at infinity?

Answer 8

the highest degree of x must be subbed out of the root and then simplified to a abs val. If infinity is being used then x will become x. If negative infinity is being used x will become -x.

Question 9

What makes a point continuous?

Answer 9

-If the function value exists.
-The limit exists
-both are equal

Question 10

How does one determine a vertical asymptote?

Answer 10

By finding the value which would make the denominator equal to 0.

Question 11

How does one determine a horizontal asymptote?

Answer 11

By determining the limit as x approaches infinity.

Question 12

What is Removable discontinuity?

Answer 12

When The limit at a point exists but does not equal the function value.

Question 13

What is Jump discontinuity?

Answer 13

When the limit at a point does not exist.

Question 14

How would one solve for variables in a piece-wise defined function if it is given that it is continuous?

Answer 14

Through a system of equations, as the limits and function value will be equal at a certain point.

Question 15

How does one determine the inverse of a function?

Answer 15

By switching the x and y and making y the subject.

Question 16

What two methods can be used when using first principles to determine the derivative?

Answer 16

• Multiplying by the LCD over the LCD to remove fractions.
• Multiplying by the conjugate over the conjugate to remove square roots.

Question 17

State the power rule formula.

Answer 17

ax^n => nax^n-1

Question 18

What makes a point differentiable?

Answer 18

-The point is continuous
-The derivative exists
-The limit of the derivative from the right is equal to the limit of the derivative from the left.

Question 19

State the chain rule.

Answer 19

(f o g)’(x) = f’(g(x)).g’(x)

Question 20

State the Product Rule.

Answer 20

d/dx[f(x).g(x)] = f’(x).g(x) + f(x).g’(x)

Question 21

State the Quotient Rule.

Answer 21

d/dx[f(x)/g(x)] = [f’(x).g(x) - f(x).g’(x)] / [g(x)]^2

Question 22

What is the most important thing not to forget when doing implicit differentiation?

Answer 22

The Chain, Product and Quotient Rules.

Question 23

How does one implicitly differentiate a function in terms of y and x?

Answer 23

-Derive both sides in terms of y and x.
-“Factorise” the dy/dx out on one side and create a fraction on the other side.

Question 24

What is the derivative of e^x

Answer 24

e^x

Question 25

What is the derivative of e^g(x)?

Answer 25

e^(g(x)) . g’(x)

Question 26

What is the derivative of a^x?

Answer 26

a^x . lna

Question 27

What is the derivative of ln x?

Answer 27

1/x

Question 28

What is the derivative of loga x?

Answer 28

1/x.ln a

Question 29

What must be done in relation to the inner function when deriving a log?

Answer 29

The derivative inner function must be multiplied to the derivative of the outer function.

Question 30

What must one always check for before deriving a term?

Answer 30

Wether or not there are any variables.