2 - 10 further differentiation

Question 1

chain rule y = f(u) and u = g(x) dy/dx =

Answer 1

dy/du *du/dx
or f’(g(x))g’(x)
eg (2x+5)^3
dy/dx = 3(2x+5)^2 *2

Question 2

y = sec x dy/dx=

Answer 2

sec x tan x

Question 3

y = cosec x dy/dx =

Answer 3

cosec x cot x

Question 4

y = cot x dy/dx =

Answer 4

-cosec^2x

Question 5

product rule y = uv

Answer 5

dy/dx = vu’ + uv’

Question 6

quotient rule y = u/v

Answer 6

dy/dx = (vu’-uv’)/v^2

Question 7

implicit differentiation eg x^2 + y^2 = 16

Answer 7

d/dx(x^2) + d/dy(y^2)*dy/dx = d/dx(16)
2x + dy/dx *2y = 0
dy/dx = -2x/2y = -x/y

Question 8

implicit differentiation d/dx(f(y))

Answer 8

= f’(y) *dy/dx

Question 9

d/dx(a^x)

Answer 9

y = a^x
ln y = x ln a
1/y dy/dx = ln a
dy/dx = y lna = a^x ln a

Question 10

differenting inverses eg use derivative of e^x to prove lnx derived = 1/x

Answer 10

dx/dy = 1/(dy/dx)
y = ln x
x = e^y
dx/dy = e^y
dy/dx = 1/e^y = 1/x