9: Differentiation

Question 1

What is the result when you differentiate sin(x)?

Answer 1

cos(x)

Question 2

What is the result when you differentiate cos(x)?

Answer 2

-sin(x)

Question 3

Differentiate sin(x) from first principles.

Answer 3

Question 4

Differentiate cos(x) from first principles.

Answer 4

Question 5

y = sin(kx)

dy/dx =

Answer 5

kcos(kx)

Question 6

y = cos(kx)

dy/dx =

Answer 6

-ksin(kx)

Question 7

y = e^kx

dy/dx =

Answer 7

ke^kx

Question 8

y = ln(x)

dy/dx =

Answer 8

1/x

Question 9

y = ln(kx)

dy/dx =

Answer 9

y = ln(kx) = ln(k) + ln(x)

ln(k) is a constant

dy/dx = 1/x

Question 10

y = a^kx

dy/dx =

Answer 10

a^kxk ln(a)

Question 11

y = a^kx and dy/dx = a^kx kln(a)

When is this true?

Answer 11

K ϵ ℝ, a > 0

Question 12

Show that the derivative of a^x is a^xln(a)

Answer 12

y = a^x

ln(y) = ln(a^x) = xln(a)

y = e^xln(a)

Where y = e^kx, dy/dx = ke^kx

dy/dx = ln(a) e^xln(a) = ln(a) e^{ln(a^x)}

= ln(a) a^x = a^x ln(a)