Pure - Differentiation

Question 1

What is the chain rule?

Answer 1

y = f(x), let f(x) = u

dy/dx = dy/du x du/dx

Question 2

What is the product rule?

Answer 2

y = f(x) g(x)

dy/dx = f(x) g’(x) + g(x) f’(x)

Question 3

What is the quotient rule?

Answer 3

y = f(x) / g(x)

dy/dx = [f’(x) g(x) - f(x) g’(x)] / [g(x)]²

Question 4

Find dy/dx of y = sin(f(x))

Answer 4

y = sin(f(x))

dy/dx = f’(x) cos(f(x))

Question 5

Find dy/dx of y = cos(f(x))

Answer 5

y = cos(f(x))

dy/dx = - f’(x) sin(f(x))

Question 6

Find dy/dx of y = Ln(x)

Answer 6

y = Ln(x)

dy/dx = 1/x

Question 7

Find dy/dx of y = Ln(f(x))

Answer 7

y = Ln(f(x))

dy/dx = f’(x) / f(x)

Question 8

Find dy/dx of y = aᵏˣ

Answer 8

y = aᵏˣ

dy/dx = kaᵏˣLn(a)

Question 9

Find dy/dx of y = [f(x)]ⁿ

Answer 9

y = [f(x)]ⁿ

dy/dx = n [f(x)]ⁿ⁻¹ f’(x)

Question 10

Find dy/dx of y = eᶠ⁽ˣ⁾

Answer 10

y = eᶠ⁽ˣ⁾

dy/dx = f’(x) eᶠ⁽ˣ⁾

Question 11

Find dy/dx of y = tan(f(x))

Answer 11

y = tan(f(x))

dy/dx = f’(x) sec²(f(x))

Question 12

Find dy/dx of y = sec(f(x))

Answer 12

y = sec(f(x))

dy/dx = f’(x) sec(f(x)) tan (f(x))

Question 13

Find dy/dx of y = cot(f(x))

Answer 13

y = cot(f(x))

dy/dx = - f’(x) cosec²(f(x))

Question 14

Find dy/dx of y = cosec(f(x))

Answer 14

y = cosec(f(x))

dy/dx = - f’(x) cosec(f(x)) cot(f(x))