Question

Substitution of a function under the root ex: 6x²/square root(3+2x³)

Answer 1

Substitution of a function under the root is:

`\[ (6x^2)/(√(3+2x^3)) \]`

Step-by-step explanation:

In order to simplify the expression
`\((6x^2)/(√(3+2x^3))\)`, we can employ the technique of substitution. Let
`\(u = 3+2x^3\)`, then the expression becomes
`\((6x^2)/(√(u))\).` Now, to further simplify, we can rewrite the expression in terms of
`\(u\).`

`\[ (6x^2)/(√(u)) = (6x^2)/(u^(1/2)) = 6x^2u^(-1/2) \]`

Next, differentiate
`\(u\) with respect to \(x\) to find \(du/dx\), which is \(6x^2\)`. Now, substitute
`\(u\) and \(du/dx\)`back into the expression:

`\[ 6x^2u^(-1/2) = (du/dx)/(u^(1/2)) \]`

Substitute
`\(3+2x^3\) back for \(u\)` to get the final expression:

`\[ (du/dx)/((3+2x^3)^(1/2)) \]`

Therefore, the simplified expression is
`\((du/dx)/(√(3+2x^3))\)`. This substitution technique helps us express the given function in a more manageable form, making it easier to analyze and work with.