Algebra of Derivative of Functions

Derivatives are a core concept in calculus used to measure the rate of change of a function. They are defined using limits and represent the instantaneous change in a quantity.

Derivatives

For a function f(x), the derivative at x = a gives the instantaneous rate of change at that point.
It is denoted by \frac{df}{dx}or f'(x). 

\frac{df}{dx} = \lim_{h \to 0}\frac{f(x + h) - f(x)}{(x + h) - (x)}

At x = a, \frac{df}{dx} = \lim_{h \to 0}\frac{f(a + h) - f(a)}{h}

Notice in the figure as the interval "h" approaches zero. The line approaches to being a tangent from a chord. This means that now the derivative when h approaches zero gives us the slope of the tangent at that particular point. 

Derivatives of some Basic Functions

The table below shows the derivatives of some standard basic functions. 

Rules of Differentiation

The table above lists derivatives of standard functions. In practice, functions are often combinations of two or more functions connected by addition, subtraction, multiplication, or division. We use following derivative rules for finding their derivatives:

• Summation or Difference Rule
• Product and Division Rule

Consider two functions f(x) and g(x). Let's say there is a third function h(x) which combines these two functions. 

Summation and Difference Rule

Case 1: h(x) = f(x) + g(x) 

This function is summation of both f(x) and g(x), the derivative of such functions is given by, 

 \frac{dh}{dx} = \frac{d}{dx}(f(x) + g(x))

⇒\frac{dh}{dx} = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))

or

h'(x) = f'(x) + g'(x) 

Case 2: h(x) = f(x) - g(x) 

This function is the difference of both f(x) and g(x), the derivative of such functions is given by, 

 \frac{dh}{dx} = \frac{d}{dx}(f(x) - g(x))

⇒\frac{dh}{dx} = \frac{d}{dx}(f(x)) - \frac{d}{dx}(g(x))

or

h'(x) = f'(x) - g'(x) 

Product and Division Rules

Case (i): h(x) = f(x) x g(x)

This function is product of both f(x) and g(x), the derivative of such functions is given by, 

 \frac{dh}{dx} = \frac{d}{dx}(f(x) \times g(x))

⇒\frac{dh}{dx} = \frac{d}{dx}(f(x))g(x) + \frac{d}{dx}(g(x))f(x)

or

h'(x) = f'(x)g(x) + g'(x) f(x)

Case (ii): h(x) = \frac{f(x)}{g(x)}

This function is division of both f(x) and g(x), the derivative of such functions is given by, 

 \frac{dh}{dx} = \frac{d}{dx}(\frac{f(x)}{g(x)})

⇒\frac{dh}{dx} = \frac{\frac{d}{dx}(f(x))g(x) - \frac{d}{dx}(g(x))f(x)}{(g(x))^2}

or

h'(x) = \frac{f'(x)g(x) - g'(x) f(x)}{(g(x))^2}

The division and product rules are also called the Leibniz rules.

Solved Examples

Problem 1: Find the derivative for f(x) = x² + 3x.

Solution: 

This function is the sum of two different function. Sum rule will be used here. 

f(x) = x² + 3x 

Here, h(x) = x² and g(x) = 3x. 

f(x) = h(x) + g(x) 

⇒f'(x) = h'(x) + g'(x) 

⇒ f'(x) = \frac{d}{dx}h(x) + \frac{d}{dx}g(x)

⇒f'(x) =\frac{d}{dx}x^2 + \frac{d}{dx}3x

⇒f'(x) = 2x + 3

Problem 2: Find the derivative for the function f(x) = e^x + sin(x).

Solution: 

This function is the sum of two different function. Sum rule will be used here. 

f(x) = e^x + sin(x) 

Here, h(x) =e^x and g(x) = sin(x)

f(x) = h(x) + g(x) 

⇒f'(x) = h'(x) + g'(x) 

⇒ f'(x) = \frac{d}{dx}h(x) + \frac{d}{dx}g(x)

⇒f'(x) =\frac{d}{dx}e^x + \frac{d}{dx}sin(x)

⇒f'(x) = e^x + cos(x)

Problem 3: Find the derivative for f(x) = 5x⁴ - 3x².

Solution: 

This function is the difference of two different function. Difference rule will be used here. 

f(x) = 5x⁴ - 3x² 

Here, h(x) =5x⁴ and g(x) = 3x² 

f(x) = h(x) - g(x) 

⇒f'(x) = h'(x) - g'(x) 

⇒ f'(x) = \frac{d}{dx}h(x) - \frac{d}{dx}g(x)

⇒f'(x) =\frac{d}{dx}5x^4 - \frac{d}{dx}3x^2

⇒f'(x) = 20x³ - 6x

Problem 4: Find the derivative for the function f(x) = 5log(x) - 3x.

Solution: 

This function is the difference of two different function. Difference rule will be used here. 

f(x) = 5log(x) - 3x

Here, h(x) =5log(x) and g(x) = 3x 

f(x) = h(x) - g(x) 

⇒f'(x) = h'(x) - g'(x) 

⇒ f'(x) = \frac{d}{dx}h(x) - \frac{d}{dx}g(x)

⇒f'(x) =\frac{d}{dx}5log(x) - \frac{d}{dx}3x

⇒f'(x) = \frac{5}{x} - 3

Problem 5: Find the derivative for f(x) = 5x⁴.sin(x)

Solution: 

This function is the product of two different function. Product rule will be used here. 

f(x) = 5x⁴.sin(x)

Here, h(x) =5x⁴ and g(x) = sin(x)

f(x) = h(x).g(x) 

⇒f'(x) = h'(x) g(x) + h(x)g'(x) 

⇒ f'(x) = \frac{d}{dx}(f(x))g(x) + \frac{d}{dx}(g(x))f(x)

⇒f'(x) =\frac{d}{dx}(5x^4)sin(x) + \frac{d}{dx}(sin(x))5x^4

⇒f'(x) = 20x³sin(x) + 5x⁴cos(x)

Problem 6: Find the derivative of f(x) = 5e^x.log(x)

Solution: 

This function is the product of two different function. Product rule will be used here. 

f(x) = 5e^x.log(x)

Here, h(x) =5e^x and g(x) = log(x)

f(x) = h(x).g(x) 

⇒f'(x) = h'(x) g(x) + h(x)g'(x) 

⇒ f'(x) = \frac{d}{dx}(f(x))g(x) + \frac{d}{dx}(g(x))f(x)

⇒f'(x) =\frac{d}{dx}(5e^x)log(x) + \frac{d}{dx}(log(x))5e^x

⇒f'(x) = 5(e^xlog(x) + \frac{1}{x}e^x)

Problem 7: Find the derivative of f(x) = \frac{x + 1}{2x}

Solution: 

This function is the division of two different function. Division rule will be used here. 

f(x)=\frac{x + 1}{2x}

Here, h(x) =x + 1 and g(x) = 2x

so,

⇒ f'(x)=\frac{g(x)h'(x) - h(x)g'(x)}{(g(x))^2}

⇒ f h'(x)=\frac{d}{dx}(x+1)=1\\g'(x) = \frac{d}{dx}(2x)=2

⇒ ff'(x)=\frac {(2x)(1)-(x+1)(2)}{(2x)^2}

Numerator:

⇒ f'(x) =2x-2(x+1)=2x-2x-2=-2

Denominator:

⇒ (2x)^2 = 4x^2

⇒ f'(x) = f'(x)=\frac{-2}{4x^2} = -\frac{1}{2x^2}

⇒ f'(x)=-\frac{1}{2x^2}

Problem 8: Find the derivative for the function f(x) = \frac{log(x)}{2x}

Solution: 

This function is the division of two different function. Division rule will be used here. 

f(x) =\frac{log(x)}{2x}

Here, h(x) =log(x) and g(x) = 2x

f(x) = \frac{h(x)}{g(x)}

⇒f'(x) = \frac{g(x)h'(x) - h(x)g'(x)}{(g(x))^2}

⇒ f'(x) = \frac{\frac{d}{dx}(h(x))g(x) - \frac{d}{dx}(g(x))h(x)}{(g(x))^2}

⇒f'(x) =\frac{\frac{d}{dx}(log(x))2x - \frac{d}{dx}(2x)(log(x))}{4x^2}

⇒f'(x) =\frac{1- 2log(x)}{4x^2}

Practice Questions

1. If f(x) = 3x² - 2x + 5 and g(x) = x³ + 4x, find the derivative of h(x) = f(x) + g(x).

2. Given f(x) = x² and g(x) = sin(x), find the derivative of their product: (f·g)(x).

3. If f(x) = e^x and g(x) = ln(x), determine the derivative of their composition: f(g(x)).

4. Find the derivative of h(x) = (x² + 1) / (x - 2) using the quotient rule.

5. If f(x) = x³ and g(x) = cos(x), find the derivative of their difference: f(x) - g(x).