Chapter 3 Equations

Question 1

dy/dx sin(x)

Answer 1

cos(x)

Question 2

dy/dx cos(x)

Answer 2

-sin(x)

Question 3

dy/dx tan(x)

Answer 3

sec^2(x)

Question 4

dy/dx sec(x)

Answer 4

sec(x)tan(x)

Question 5

dy/dx csc(x)

Answer 5

-csc(x)cot(x)

Question 6

dy/dx cot(x)

Answer 6

-csc^2(x)

Question 7

dy/dx sin^-1(x)

Answer 7

1 / sqrt(1-x^2)

Question 8

dy/dx cos^-1(x)

Answer 8

-1 / sqrt(1-x^2)

Question 9

dy/dx tan^-1(x)

Answer 9

1 / (1+x^2)

Question 10

dy/dx cot^-1(x)

Answer 10

-1 / (1+x^2)

Question 11

dy/dx csc^-1(x)

Answer 11

-1 / (abs(x) * sqrt(x^2 - 1))

Question 12

dy/dx sec^-1(x)

Answer 12

1 / (abs(x) * sqrt(x^2 - 1))

Question 13

dy/dx logx (base a)

Answer 13

1/(x*lna)

Question 14

dy/dx a^x

Answer 14

a^x * lna

Question 15

dy/dx e^x

Answer 15

e^x

Question 16

d/dx a^u(x)

Answer 16

a^u(x) * ln(a) * u’(x)

f(x)=a^u(x)
ln(f(x))=u(x)ln(a) #take the natural log of both sides, rearrange
(1/f(x))f’(x)=ln(a)u’(x) #ln(a) is a constant
f’(x)=f(x)ln(a)u’(x) #we know f(x)
f’(x)=(a^u(x))ln(a)*u’(x) #substitute for f(x). Done

Question 17

f(x) and g are inverse functions. What is the slope of g(x)?

Answer 17

g’(y0)=1/f’(x0)

example, f(x)=2x
f(2)=4, so g(4)=2
g’(4)=1/f’(g(4))=1/f’(2)=1/2

Question 18

derivative of f(g(x))

Answer 18

f’(g(x))*g’(x)

Leibniz notation: dy/dx=dy/du * du/dx

Question 19

d/dx f(x)^n

Answer 19

n*f(x)^(n-1) * f’(x)

Question 20

Difference between implicit and explicit?

Answer 20

implicit: y is not isolated
explicit: y is isolated

Question 21

arc before a trig function

Answer 21

inverse. ex: sin^-1 (x) = arcsin(x)