ch9 Differentiation

Question 1

How do you show that a tangent to a point does not intersect the curve again?

Answer 1

Substitute x and y in the equation of the normal from part b. You should then get two possible values of for sint, and one of them is invalid for the given domain.

With the valid solution show that it gives another value of T that causes the same point at A

Question 2

What is important when differentiating X and Ys of the same function?

Answer 2

Differentiate the y normally then multiply it by dy/dx.

Then make dy/dx the subject.

Question 3

What to do when there is an implicit differentiation question but there is a fraction involved?

Answer 3

Multiply through by the denominator of the fraction to eliminate it.

Question 4

What to do when implicitly differentiating and then the dy/dx is trapped in a bracket?

Answer 4

Expand it by the product of the other 2 terms and then add or subtract this from the dy/dx that is not trapped then the sum of this will be (1+ The coefficient of the trapped dy/dx that was expanded).

Question 5

What to do when there is a single constant to the power of x?

Answer 5

Take natural logs of both sides and then differentiate implicitly.

Question 6

What to do when you have two terms in a bracket beside a log and you must differentiate implicitly?

Answer 6

multiply both terms In the bracket by the log.

Question 7

How to show that there are no points on the curve such that dy/dx=0

Answer 7

First differentiate.
Then equal dy/dx to 0.
Then create an equation with y as the subject.
Sub this into original equation
Show that discriminant of original discriminant is less than 0

Question 8

What is the most important aspect of differentiating Sin and cos x from first principles?

Answer 8

SMALL ANGLE APPROXIMATIONS.

cosh-1/h tends to 0

sinh/h tends to 1

Question 9

What is the rule for differentiating y=e^x

Answer 9

stays as e^x

Question 10

What is the rule for differentiating y=e^kx

Answer 10

derivative of kx becomes the coefficient of e^x

dy/dx = ke^kx

Question 11

How to differentiate ln(f(x))

Answer 11

f’(x)/f(x).

Differential of f(x) divided by f(x)

Question 12

How to prove the differentiating ln(x)

Answer 12

y=ln(x)

inverse log both sides

e^y=x

dx/dy = e^y

dy/dx = 1/ e^y

sub in e^y=x

Question 13

What is the rule for differentiating y=a^x

Answer 13

dy/dx = a^x x ln(a)

Question 14

How do you prove the differential of y=a^x

Answer 14

step 1 - take natural logs of both sides. ln(y)=ln(a^x)
step 2 -use power law to bring x down ln(y)=xln(a)
step 3 - e both sides y = e^xlna
Step 4 - now differentiate y = lna x e^xlna
step 5 sub in y= e^xlna so y= lna x y
step 5 sub in original y=a^x
step 6 y= lna x a^x

Question 15

What is the chain rule?

Answer 15

Differentiate the inside first
Differentiate the outside second.

Derivative for the bracket x power x bracket to the power of one less.

Question 16

When is the product rule used?

What is the product rule?

Answer 16

• used when there are two functions.

- v du/dx + u dv/dx

Question 17

When is the quotient rule used?

what is the quotient rule?

Answer 17

-a fraction, f(x)/g(x)

f’(x)g(x) - g’(x)f(x) / (g(x))^2

Question 18

what is the derivative of tan(x)?

and how do you prove it?

Answer 18

tan(x) diffs to sec^2(x)
Swap tan(x) to sin(x)/cos(x) then use quotient rule from there.

Question 19

What is the differential of cosec(x)?

How would you prove it?

Answer 19

• cosec(x)cot(x)

- use quotient rule for 1/sin(x)

Question 20

What is the differential of sec(x)?

Answer 20

sec(x)tan(x)

Question 21

What is the differential of cot(x)?

Answer 21

-cosec^2(x)

Question 22

What is the difference between a concave and convex function?

What is a way to remember this?

Answer 22

Concave function, line segment joining two points is below function Eg n shape.

Convex function, line segment joining two points is above function Eg U shape.

Concave is underground.

Question 23

How to identify is a function is concave or convex?

Answer 23

convex d^2y/dx^2 greater than or equaled to zero.

concave d^2y/dx^2 less than or equaled to zero.

Question 24

What is a point of inflexion?

What is a calculus property of this?

Answer 24

when the curve changes from concave to convex.

Second differential= 0

Question 25

When showing that a curve is convex in a given interval with 6x-4 what is important to mention?

Answer 25

-That it is a linear relationship.

Question 26

What does a rate of change mean?

Answer 26

That the differential with have a dt on the bottom.

Question 27

What would be the differential if it says the rate at which particles decay are proportional to K?

Answer 27

dP/Dt= -K