Determine the derivative of the following (3 {x}^(2) - √(x) ) {}^(2) ​

Question

Determine the derivative of the following


`(3 {x}^(2) - √(x) ) {}^(2)`
​

Answer 1

`\huge{ \boxed{36 {x}^(3) - \frac{6 {x}^(2) }{ √(x) } - 12x √(x) + 2 √(x) }}`

Explanation:

To find the derivative of the function
`(3 {x}^(2) - √(x) )^(2)` , the chain rule can be applied.

• First let's define the function as
  `f(x) = (3 {x}^(2) - √(x) )^(2)`
• Next, the function can be rewritten as f(x) = u(x)² , where
  `u(x) = (3 {x}^(2) - √(x))`

According to the chain rule, the derivative of f(x) with respect to x is given by:

`(df)/(dx) = (df)/(du) * \: (du)/(dx)`

where

`(df)/(du)` represents the derivative f(x) with respect to u.

• Since f(x) = u(x)² , it can be differentiated as:

`(df)/(du) = 2u(x)`

• Next we differentiate
  `u(x) = (3 {x}^(2) - √(x))` which is given as:

`(du)/(dx) = (d)/(dx) ( {3x}^(2) ) - (d)/(dx) ( √(x) ) \\ \\ (du)/(dx) = {6x} - (1)/(2 √(x) )`

• Next we substitute the calculated values back into the chain rule formula given above, we have:

`(df)/(dx) = 2u(x) * (6x - (1)/(2 √(x) ) ) \\`

• Lastly we substitute
  `u(x) = (3 {x}^(2) - √(x))` back into the equation and simplify:

`(df)/(dx) = 2(3 {x}^(2) - √(x) ) * (6x - (1)/(2 √(x) ) ) \\ = (6 {x}^(2) - 2 √(x) )( 6x - (1)/(2 √(x) ) ) \\ = 36 {x}^(3) - \frac{6 {x}^(2) }{ √(x) } - 12x √(x) + 2 √(x)`

We have the final answer as

`36 {x}^(3) - \frac{6 {x}^(2) }{ √(x) } - 12x √(x) + 2 √(x)`