Question

Differentiatey = 3x√x⁴-5

Answer 1

Given:

`y=3x√(x^4-5)`

Required:

We need to differentiate the given expression.

Step-by-step explanation:

Consider the given expression.

`y=3x√(x^4-5)`
`y=3x(x^4-5)^{(1)/(2)}`

Differentiate the given expression with respect to x.

`Use\text{ }(uv)^(\prime)=uv^(\prime)+vu^(\prime).\text{ Here u=3x and v=}(x^4-5)^{(1)/(2)}.`
`y^(\prime)=3x((1)/(2))(x^4-5)^{(1)/(2)-1}(4x^3)+(x^4-5)^{(1)/(2)}(3)`
`y^(\prime)=\frac{3x(4x^3)}{2\left(x^4-5\right)^{(1)/(2)}}+3(x^4-5)^{(1)/(2)}`
`y^(\prime)=\frac{6x^4}{\left(x^4-5\right)^{(1)/(2)}}+3(x^4-5)^{(1)/(2)}`
`y^(\prime)=\frac{6x^4}{\left(x^4-5\right)^{(1)/(2)}}+\frac{3(x^4-5)^{(1)/(2)}(x^4-5)^{(1)/(2)}}{(x^4-5)^{(1)/(2)}}`
`y^(\prime)=\frac{6x^4}{\left(x^4-5\right)^{(1)/(2)}}+\frac{3(x^4-5)}{(x^4-5)^{(1)/(2)}}`
`y^(\prime)=\frac{6x^4}{\left(x^4-5\right)^{(1)/(2)}}+\frac{3x^4-15}{(x^4-5)^{(1)/(2)}}`
`y^(\prime)=\frac{6x^4+3x^4-15}{\left(x^4-5\right)^{(1)/(2)}}`
`y^(\prime)=\frac{9x^4-15}{\left(x^4-5\right)^{(1)/(2)}}`
`y^(\prime)=\frac{3(3x^4-5)}{\left(x^4-5\right)^{(1)/(2)}}`
`y^(\prime)=(3(3x^4-5))/(√(x^4-5))`
`y^(\prime)=(3(3x^4-5))/(√(x^4-5))*(√(x^4-5))/(√(x^4-5))`
`y^(\prime)=(3(3x^4-5)√(x^4-5))/(x^4-5)`

Final answer:

`y^(\prime)=(3(3x^4-5)√(x^4-5))/(x^4-5)`