5. The derivative of the sum, difference, quotient, and product of functions

The derivative of the sum, difference, quotient, and product of functions

Example: Find derivative of y= x²+3x-5

Ans:

=2x +3

Example: Find derivative of y=( x²+3x)(2x-7)

Ans:

Example: Find derivative of y=( x²+3x)/(2x-7)

Example : Find the derivative of sin x – cos x.

Solution: 

Given function is: sin x – cos x

Let f(x) = sin x and g(x) = cos x

Using the difference rule of differentiation,

d/dx [f(x) – g(x)] = d/dx f(x) – d/dx g(x)

d/dx (sin x – cos x) = d/dx (sin x) – d/dx (cos x)

= cos x – (-sin x)

= cos x + sin x

Example : Find the derivative of (5x³ – 3x + 1)(x + 1).

Solution: 

Given: (5x³ – 3x + 1)(x + 1)

Let f(x) = (5x³ – 3x + 1) and g(x) = (x + 1)

Using the product rule of differentiation,

d/dx [f(x) .g(x)] = f(x) [d/dx g(x)] + g(x) [d/dx f(x)]

= (5x³ – 3x + 1) [d/dx (x + 1)] + (x + 1) [d/dx (5x³ – 3x + 1)]

= (5x³ – 3x + 1) (1 + 0) + (x + 1)[5(3x²) – 3(1) + 0]

= (5x³ – 3x + 1) + (x + 1)(15×2 – 3)

= 5x³ – 3x + 1 + 15x³ – 3x + 15x² – 3

= 20x³ + 15x² – 6x – 2