Rules of Differentiation – SS2 Further Mathematics Lesson Note (Third Term)

Differentiation finds the derivative dy/dx of y with respect to x. This note gives four sets of rules: differentiating a sum term by term, the standard derivatives of trigonometric, logarithmic and exponential functions, the chain rule for a function of a function, and the product rule for two functions multiplied together. Each rule has worked examples.

Behavioural objectives

By the end of the lesson, students should be able to:

• differentiate a sum of terms term by term;
• state and use the derivatives of sin x, cos x, tan x, cot x, sec x, ln x and eˣ;
• differentiate functions such as sin kx and e^kx, where k is a constant;
• apply the chain rule to a function of a function;
• apply the product rule to the product of two functions of x.

How do you differentiate a sum of functions?

If y = u(x) + v(x), where u and v are functions of x, then:

dy/dx = du/dx + dv/dx

Example 1

Find the derivative of y = 3x⁵ – 2x⁴ – 2x²

Solution

y = 3x⁵ – 2x⁴ – 2x²

dy/dx = 5 × 3x⁵⁻¹ – 4 × 2x⁴⁻¹ – 2 × 2x²⁻¹

dy/dx = 15x⁴ – 8x³ – 4x

Example 2

Find the derivative of y = 3x⁵ – 2x⁴ + 6x³ + 3x – 11

Solution

y = 3x⁵ – 2x⁴ + 6x³ + 3x – 11

dy/dx = 5 × 3x⁵⁻¹ – 4 × 2x⁴⁻¹ + 3 × 6x³⁻¹ + 1 × 3x¹⁻¹ – 0 × 11x⁰

dy/dx = 15x⁴ – 8x³ + 18x² + 3

How do you differentiate trigonometric and other transcendental functions?

Standard results

When the angle is multiplied by a constant

If the angle is multiplied by a constant k, then:

Example

Differentiate y = 3e⁵ˣ

Solution

y = 3e⁵ˣ

dy/dx = 3 × 5e⁵ˣ

dy/dx = 15e⁵ˣ

What is the function of a function (chain) rule?

Suppose that y is a function of u and u is a function of x. In other words, if y = f(u) and u = g(x), then:

dy/dx = (dy/du) × (du/dx)

This is called the chain rule for differentiation.

Example 1

Find the derivative of y = (x⁷ – 12)²

Solution

Let u = x⁷ – 12, then y = u²

So, dy/du = 2u and du/dx = 7x⁶

Hence, dy/dx = (dy/du) × (du/dx)

= 2(x⁷ – 12) × 7x⁶

= 14x⁶(x⁷ – 12)

Example 2

Find the derivative of y = sin 6x

Solution

Let u = 6x, then y = sin u

Therefore, dy/du = cos u and du/dx = 6

Hence, dy/dx = (dy/du) × (du/dx)

= cos u × 6

= 6 cos 6x

What is the product rule?

If y = uv, where u and v are functions of x, then:

dy/dx = u(dv/dx) + v(du/dx)

Example

Find the derivative of y = x² sin 6x.

Solution

y = x² sin 6x

Let u = x² and v = sin 6x. Then:

du/dx = 2x
dv/dx = 6 cos 6x

Therefore,

dy/dx = u(dv/dx) + v(du/dx)

= x²(6 cos 6x) + sin 6x(2x)

= 6x² cos 6x + 2x sin 6x

Evaluation

1. Find the derivative of y = 4x³ – 5x² + 7x – 9.
2. Differentiate y = 2 cos 3x.
3. Use the chain rule to differentiate y = (x³ + 5)⁴.
4. Use the product rule to differentiate y = x³ cos 2x.

Assignment

1. Find the derivative of y = 6x⁴ + 2x³ – x + 8.
2. Differentiate y = 5e⁴ˣ.
3. Use the chain rule to differentiate y = (3x² – 1)³.
4. Use the product rule to differentiate y = x sin 5x.