Differentiation

Question 1

Differentiate:

y = (f(x))^n

when f(x) contains multiple terms

(The Chain Rule)

Answer 1

dy/dx = n x f’(x) x (f(x))^(n-1)

Question 2

Differentiate:

y = e^f(x)

Answer 2

dy/dx = f’(x)e^f(x)

Question 3

Differentiate:

y = a^x

when a does not equal 1

Answer 3

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

Question 4

Differentiate:

y = ln(kx)

Answer 4

dy/dx = 1/x

Question 5

Diffentiate:

y = ln(f(x))

Answer 5

dy/dx = f’(x) / f(x)

Question 6

Differentiate:

y = sin(kx)

Answer 6

dy/dx= kcos(kx)

Question 7

Differentiate:

y = cos(kx)

Answer 7

dy/dx= -ksin(kx)

Question 8

Differentiate:

y = tan(kx)

Answer 8

dy/dx = ksec^2(kx)

Question 9

Differentiate

y = u x v

(Product Rule)

Answer 9

dy/dx = u x dv/dx + v x du/dx

Question 10

Differentiate:

y = u/v

(Quotient Rule)

Answer 10

dy/dx = ((v x du/dx)-(u x dv/dx)) / v^2

Question 11

dx/dy =

Answer 11

= 1/ (dy/dx)

(and vice versa)

Question 12

Implicit Differentiation

when differentiating y^(n) with respects to x, you get….

Answer 12

ny^(n-1) x dy/dx

(NEVER USE THE QUOTIENT RULE DURING IMPLICIT DIFFERENTIATION)

Question 13

Differentiate:

y= a^f(x)

Answer 13

dy/dx = ln(a) x a^f(x) x f’(x)

Question 14

Concave curve

Answer 14

d2y/dx2 < 0

As gradient is decreasing

Question 15

Convex Curve

Answer 15

d2y/dx2 >0

As gradient is increasing

Question 16

Draw:

A concave curve

Answer 16

Refer to notes for answers

Question 17

Draw:

A convex curve

Answer 17

Refer to notes for answers

Question 18

Differentiate

sinx using first principles

Answer 18

refer to notes for answers

Question 19

Differentiate

cosx using first principles

Answer 19

Refer to notes for answer