Differentiation

Question 1

What does lnx differentiate to?

Answer 1

1/x

Question 2

When a function is increasing, what does that mean?

Answer 2

dy/dx>0, gradient is positive

Question 3

When a function is decreasing, what does that mean?

Answer 3

dy/dx<0, gradient is negative

Question 4

How to differentiate natural log functions

Answer 4

e^x = e^x
e^kx = ke^kx (power never changes basically)

Question 5

cosx differentiates to

Answer 5

-sinx

Question 6

-sinx differentiates to

Answer 6

-cosx

Question 7

-cosx differentiates to

Answer 7

sinx

Question 8

sinx differentiates to

Answer 8

cosx

Question 9

lnkx+c differentiates to

Answer 9

k/kx+c

Question 10

a^x differentiates to

Answer 10

a^xlna, by taking natural logs and implicit differentiation

Question 11

a^kx differentiates to

Answer 11

ka^kxlna

Question 12

sin(kx) differentiates to

Answer 12

kcos(kx)

Question 13

cos(kx) differentiates to

Answer 13

-ksin(kx)

Question 14

product rule

Answer 14

u(dv/dx) + v(du/dx)

Question 15

differentiate arcsinx

Answer 15

1/(√1-x^2)

Question 16

differentiate arccosx

Answer 16

-(1/(√1-x^2))

Question 17

differentiate arctanx

Answer 17

1/(1+x^2)

Question 18

How do you differentiate arc trig functions?

Answer 18

use the rule dy/dx is reciprocal of dx/dy
if y = arctrigx, then x = trigy

Question 19

How to differentiate parametric functions?

Answer 19

Use (dy/dt)/(dx/dt) = dy/dx. differentiate each separately with respect to t.

Question 20

How to test whether a point is a point of inflection?

Answer 20

See if second differential = 0, then see if sign changes on second differential either side of test point.

Question 21

How to implicitly differentiate xy?

Answer 21

Use product rule.

Question 22

How do you differentiate sin^2x?

Answer 22

Imagine it as (sinx)^2 and use chain rule.

Question 23

If a function is concave, what does that mean?

Answer 23

Second derivative is positive

Question 24

If a function is convex, what does that mean?

Answer 24

Second derivative is negative

Question 25

What does ln2x differentiate to?

Answer 25

1/x