Chapter 9: Differentiation

Question 1

dy/dx of y = sin(kx)
dy/dx of y = cos(kx)

Answer 1

dy/dx = kcos(kx)
dy/dx = -ksin(kx)

Question 2

dy/dx of y = e^kx

Answer 2

dy/dx = ke^kx

Question 3

dy/dx of y = ln(x)

Answer 3

dy/dx = 1/x

Question 4

dy/dx of y = a^(kx)

Answer 4

dy/dx = a^(kx) k ln(a)

Question 5

The chain rule

Answer 5

dy/dx = dy/du X du/dx

Where y is a function of u and u is a function of x

Question 6

The product rule

Answer 6

dy/dx = uv’ + vu’

If y = uv
Where u and v are functions of x

Question 7

The quotient rule

Answer 7

dy/dx = (vu’ - uv’)/v^2

If y = u/v
Where u and v are functions of x

Question 8

dy/dx of y = tan(kx)

Answer 8

dy/dx = k sec^2 (kx)

Question 9

dy/dx of y = cosec(kx)

Answer 9

dy/dx = -k cosec(kx) cot(kx)

Question 10

dy/dx of sec(kx)

Answer 10

dy/dx = k sec(kx) tan(kx)

Question 11

dy/dx of y = cot(kx)

Answer 11

dy/dx = -k cosec^2 (kx)

Question 12

Parametric differentiation

Answer 12

dy/dx =(dy/dt)/(dx/dt)

Question 13

Implicit differentiation

Answer 13

d/dx (y^n) = ny^(n-1) dy/dx

d/dx (xy) = x dy/dx + y

Question 14

Using second derivatives

Answer 14

Concave if f’’(x) < 0
Convex if f’’(x) > 0
Point of inflection if f‘’(x) changes sign

Question 15

Rates of change

Answer 15

dA/dt = dA/dx X dx/dt