Differentiation

Question 1

Basic differentiation

Answer 1

y = axⁿ
y’ = n x axⁿ⁻¹

Question 2

Differentiation of the exponential function

Answer 2

This means that if f(x) = eˣ then f’(x) = eˣ

This means if f(x) = eᵍ⁽ˣ⁾ then
f’(x) = g’(x) x eᵍ⁽ˣ⁾

Question 3

Chain rule for differentiation of the exponential function

Answer 3

Chain rule:
y(u(x))
y’ = dy/dx = dy/du . du/dx

Question 4

Tangent to curves

Answer 4

Step:
1. Find the gradient function f(x) → f’(x)
2. Substitute x₁ into the derived function to find the tangent gradient f’(x₁) = m
3. Substitute the gradient m and the point (x₁, y₁) into the equation of the tangent line

y – y₁ = m (x – x₁)
x₁ = a
y₁ = b
y – b = m(x – a)

Question 5

Differentiation of log function

Answer 5

If y = 𝑙𝑛|𝑢(𝑥)| then 𝑦′ = 𝑢′(𝑥) ×[1/𝑢(𝑥)]
, 𝑢(𝑥) > 0

Question 6

Normal to curves

Answer 6

The grad of the normal is the negative reciprocal

Steps:
1. Find the gradient function f(x) → f’(x)
2. Substitute x₁ into the derived function to find the tangent gradient f’(x₁) = m
3. Substitute the gradient of tangent m into m of normal
4. Sub the new gradient and the point (x₁, y₁) into the equation of the tangent line

y – y₁ = m (x – x₁)
x₁ = a
y₁ = b
y – b = m(x – a)

Question 7

Finding stationary points

Answer 7

Differentiate the function to find f(x)

Place f’(x) = 0 and solve for x

Sub the value into the original function to find the y-value of the stationary point

Determine nature of point by taking the second derivative

y’’ > 0 (min)
y’’ < 0 (max)

Question 8

Differentiation of exponential function with log

Answer 8

If f(x) = bˣ then f’(x) = ln(b) x bˣ

Question 9

Power rule

Answer 9

f(x) = [u(x)]ⁿ
f’(x) = n . [u(x)]ⁿ⁻¹ . du/dx

Question 10

Differentiating products

Answer 10

y = uv

When 2 separate functions with x in them are multiplying each other, we use the product rule. One product is u and the other is v

y = u’v + v’u

Question 11

Differentiating quotients

Answer 11

y = u / v

To use the quotient rule we need one function divided by another, the one at the top is u and the bottom one is v

y = [u’v - v’u] / v²

Question 12

Parametric equations

Answer 12

x = p + r cos(θ)
y = q + r sin(θ)

Question 13

Implicit differentiation

Answer 13

any y value must have dy/dx multiplied by it once differentiated

REMEMBER PRODUCT RULE FOR COMBINED TERMS