CHAMPION TUTORS
Friday, 17th September, 2021.
DIFFERENTIAL CALCULUS (DIFFERENTIATION)
CALCULUS (on a broad scale) is the branch of mathematics that deals with the finding and properties
of derivatives and integrals of functions.
(Sounds complex right? It’s not at all)
What is differentiation?
Differentiation is a branch of calculus that is applied in finding the slope or gradient of a
line, maximum and minimum points attained by a function, velocity and acceleration of a
particle etc
To get the derivative of a function
Suppose you are given a function y = ax^n, the derivative of the function called dy/dx
(pronounced ‘dee y, dee x'), is = nax^n-1
(NB: dy/dx means: differentiate y with respect to x)
This is what the above means :
To get the derivative of any function:
I. Use the power of the variable to multiply the expression
II. Less the power of the variable by 1
Let’s put this to test:
Example 1
If y = 3x², find the derivative of the function y
Solution
y = 3x²
The variable in 3x² is x and the power of x is 2
So, let’s see the steps:
(Use the power of the variable to multiply the expression)
So we have:
2 × 3x²
= 6x²
(Less the power of the variable by 1)
So we have:
So, the derivative (dy/dx) = 6x
Example 2
If y = 7x⁵, find the derivative of the function y
Solution
y = 7x⁵
The variable is x and its power is 5
(Use the power of the variable to multiply the expression)
= 5 × 7x⁵
= 35x⁵
(Less the power of the variable by 1)
So, dy/dx = 35x⁴
Example 3
What is the derivative of the expression 6x^-2?
Solution
Example 4:
If u = 6x², what is the derivative of the function u?
Solution
Note that if u = 6x², the derivative will be called du/dx because you are differentiating u
with respect to x. Saying dy/dx in such a case will be incorrect
u = 6x²
(Use power to multiply expression)
= 2 × 6x²
= 12x²
(Less power by 1)
So du/dx = 12x
Example 5
Solution
Note again that if v = 7x^-4 / 2, the derivative will be called dv/dx
If you call it dy/dx, you will be incorrect
Example 6
If t = 12/x³, find the derivative of t
Solution
The derivative of t will be called dt/dx
Example 7
Find the derivative of √x
Solution
Finding the derivative of a binomial, trinomial, etc expression
(A binomial is an algebraic expression with two (bi) terms. For example, 2n + 5, 6x – 1, 7x² +
5 etc
A trinomial is an algebraic expression with three (tri) terms. For example, 2x² - 3 + 4, 5x⁴ -
3x³ + 5, 7x⁶ + 7x⁴ - 2x² etc)
The derivative of a binomial, trinomial etc is the sum of the derivative of each individual
term
To get the derivative of such:
• Find the derivative of each term
• Sum up the derivative
Let’s see how we use these steps now
Example 1
Find the derivative of 4x² + 3x
Solution
Let y = 4x² + 3x
The derivative here will be called dy/dx
The first term is 4x² and the second term is +3x
The derivative of 4x² = 8x
The derivative of 3x = 3
So dy/dx = 8x + 3
Example 2:
Find the derivative of 5x³ - 2x² + 2
Solution
Let t = 5x³ - 2x² + 2
The derivative will be called dt/dx
The first term = 5x³. The derivative of 5x³ = 15x²
The second term = -2x². The derivative of -2x² = -4x
The third term = 2. The derivative of 2 = 0
(The derivative of any constant/number = 0)
So, dt/dx = 15x² – 4x + 0 = 15x² - 4x
Example 3
If h = r² - r + 3, find dh/dr
Solution
dh/dr is the derivative, so we are to find the derivative of r² - r + 3
The first term is r². The derivative of r² = 2r
The second term is -r. The derivative of -r = -1
The third term is +3. The derivative of +3 = 0
So, the dh/dr = 2r – 1 + 0
= 2r – 1
Example 4
y = 5/x³ + 6/x² - 5x + 1, find dy/dx
Solution
The derivative is dy/dx
Example 5
If y = 3√x – 10/x³
Solution
y = 3√x – 10/x³
Finding the derivative of a binomial, trinomial etc with a power
A binomial, as you have previously learnt, is an algebraic expression with 2 terms. When it
has a power, it looks something like (2x + 3)² or (4x² - 3)³ or (5x – 1)⁷ etc
A trinomial, as you have previously learnt as well, is an algebraic expression with 3 terms.
When it has a power, it looks something like (5x² + 5x – 3)² or (5x³ - 5x² - x)⁴ etc
Take for example, you are given a question like this: Find the derivative of (2x + 3)², you may
be so wise as to expand it, that is:
(2x + 3)² = (2x + 3)(2x + 3)
(2x + 3)² = 4x² + 6x + 6x + 9
(2x + 3)² = 4x² + 12x + 9
Then you’ll say to yourself, “Finding the derivative of (2x + 3)² is the same as Finding the
derivative of 4x² + 12x + 9, so I’ll just find the derivative of 4x² + 12x + 9”
The derivative of 4x² + 12x + 9 = 8x + 12
You will be very correct if you do so!
But what if you were asked to find the derivative of (3x + 5)⁸. You might sleep on this if you
attempt to expand it
So, let’s see how we can find the derivative of a binomial, trinomial etc with a power.
To get the derivative of a binomial, trinomial etc with a power:
• Replace what is inside the bracket with ‘u’
• Find the derivative of the resulting expression
• Multiply that derivative with the derivative of what was initially in the bracket
• Replace ‘u' with what was initially in the bracket.
Let’s see how these steps work in a real life example
Example 1
What is the derivative of (3x + 3)⁴?
Solution
Let y = (3x + 3)⁴
• Replace what is inside the bracket with ‘u’
That gives us y = u⁴
• Find the derivative of the resulting expression
u⁴ is the resulting expression
The derivative of u⁴ = 4u³
• Multiply that derivative with the derivative of what was initially in the bracket
3x + 3 was initially in the bracket
The derivative of 3x + 3 = 3
So we are multiplying 4u³ by 3
That gives us 12u³
• Replace ‘u' with what was initially in the bracket.
12u³
= 12(3x + 3)³
So the derivative dy/dx = 12(3x + 3)³
Example 2
If h = (x³ - 2x²)³, find dh/dx.
Solution
dh/dx is the derivative
• Replace what is inside the bracket with ‘u’
= u³
• Find the derivative of the resulting expression
u³
The derivative of u³ = 3u²
• Multiply that derivative with the derivative of what was initially in the bracket
x³ - 2x² was initially in the bracket
The derivative of x³ - 3x² = 3x² - 4x
So we are multiplying 3u² by 3x² - 4x
That gives us u² × 9x² - 12x
• Replace ‘u' with what was initially in the bracket.
u² × 9x² - 12x
= (x³ - 2x²)² × (9x² - 12x)
= (9x² - 12x)(x³ - 2x²)²
(Note: You can expand it if you choose to depending on how the answer choice may look
like. It may require you to expand it to see it)
Example 3
Find the derivative of (3x + 5)⁴
Solution
• Replace what is inside the bracket with ‘u’
u⁴
• Find the derivative of the resulting expression
u⁴ is the resulting expression
The derivative of u⁴ = 4u³
• Multiply that derivative with the derivative of what was initially in the bracket
3x + 5 was initially in the bracket
The derivative of 3x + 5 = 3
So we multiply 3 by 4u³
= 12u³
• Replace ‘u' with what was initially in the bracket.
12u³
= 12(3x + 5)³
Example 4
If y = (2x³ - x²)⁵, find dy/dx
Solution
dy/dx is the derivative.
• Replace what is inside the bracket with ‘u’
u⁵
• Find the derivative of the resulting expression
The derivative of u⁵ = 5u⁴
• Multiply that derivative with the derivative of what was initially in the bracket
2x³ - x² was initially in the bracket
The derivative of 2x³ - x² = 6x² - 2x
So we multiply 5u⁴ by 6x² - 2x
= 5u⁴(6x² - 2x)
• Replace ‘u' with what was initially in the bracket.
5u⁴(6x² - 2x)
= 5(2x³ - x²)⁴ (6x² - 2x)
GETTING THE VALUE OF THE DERIVATIVE OF A FUNCTION at a point
When you are asked to get the derivative at a point,
• Get the derivative
• Substitute the value of x in that point in the derivative √x
EXAMPLE 1
If y = 3x², find the derivative of the function y at point x = 1
Solution
The derivative of 3x² = (2 × 3x¹)
= 6x
Hence, the derivative at point x = 1
= 6(1)
=6
Example 2
Find the derivative of 4x² + 3x at point x = -2
Solution
Let y = 4x² + 3x¹
Then, dy/dx = (2 × 4x¹) + (1 × 3x⁰)
= 8x + 3
So the derivative at x = -2
= 8(-2) + 3
= -16 + 3
= -13
Example 3
If y = 3√x – 10/x³, find dy/dx at x = 1
Solution
y = 3√x – 10/x³
So, dy/dx at x = 1,
= 3/2√1 + 30/1⁴
= 3/2 + 30/1
= 3/2 + 30
= 1.5 + 30
= 31.5
= 63/2
