Chapter 22 - Differentiation 2 (Pure)

Question 1

what are convex curves?

Answer 1

convex curves are u shaped so has an increasing gradient and in turn a positive second derivative

Question 2

what is a convex curve?

Answer 2

a convex curve looks like an n shape and has a decreasing gradient and in turn a negative second derivative

Question 3

what are points of inflection?

Answer 3

a point where a curve changes from concave to convex.

remember f’‘(x)=0 at a point of inflection

Question 4

what’s the chain rule?

Answer 4

y = f(g(x))

dy/dx = g⁻¹ (x) f⁻¹(g(x))

Question 5

what’s the connected rate of change rule?

Answer 5

dy/dx = dy / du X du / dx

Question 6

when is the chain rule used?

Answer 6

the chain rule is used to differentiate composite functions
like f(g(x))

Question 7

how to differentiation Y= e ^f(x)

Answer 7

Y= e ^f(x) dy/dx = f’(x) e ^ f(x)

Question 8

how to differentiate Y= ln(x)

Answer 8

Y= ln(x) dy/dx = 1/x

or

dy/dx = f’(x) / f(x)

Question 9

what do cos and sin x differentiate to?

Answer 9

sin x
cos x
-sin x
- cos x

differentiate down integrate up

btw its a loop :)

Question 10

what does tan differentiate to?

Answer 10

tan differentiates to sec² x

Question 11

what’s the product rule? when do you use the product rule?

Answer 11

if Y=uv, dy/dx= u dv/dx + v du/dx

where u and v are functions

Question 12

when to use the quotient rule?

Answer 12

you use the quotient rule when its a fraction

Question 13

how to differentiate parametric equations? what is the equation?

Answer 13

dy/dx = dy/dt ÷dx/dt

Question 14

what is implicit differentiation?

Answer 14

‘implicit differentiation’ is used when an equation is written in the form f(x,y) = g(x,y)

example: y²= xy + x + 2 is implicit

Question 15

how to do implicit differentiation?

Answer 15

step 1 : differentiate terms in the x only with respect to x
step 2: use the chain rule to differentiate the y only
step 3: use the the product rule to differentiate terms in both x and y
step 4: re arrange to make dy/dx the subject

Question 16

what’s y= arcsin x differentiate?

Answer 16

y= arcsin x

dy/dx = 1/ √ 1 -x²

Question 17

what’s y= arccos x differentiate to?

Answer 17

y= arccos x

dy/dx = - 1 / √1-x²