Question

Differentiate with respect to x : 2xe^x​

Answer 1

`{ \tt{y = 2 {xe}^(x) }}`

» We are going to use product rule of ∂

`{ \boxed{ \tt{ \: (dy)/(dx) = { \huge{ \red{v}}} (du)/(dx) } + { \huge{ \green{u}}} (dv)/(dx) }}`

• v is e^x
• u is 2x
• du/dx is 2
• dv/dx is e^x

`{ \tt{ (dy)/(dx) = ( {e}^(x) )(2) + (2x)( {e}^(x) )}} \\ \\ { \tt{ (dy)/(dx) = 2 {e}^(x) + 2x {e}^(x) }} \\ \\ { \boxed{ \rm{ \: (dy)/(dx) = 2 {e}^(x)(1 + x) }}}`

Answer 2

`\\ \sf\longmapsto (d)/(dx)2xe^x`

• Apply product rule

`\\ \sf\longmapsto (d)/(dx)2x(e^x)`

`\\ \sf\longmapsto 2x(d)/(dx)e^x+e^x(d)/(dx)2x`

`\\ \sf\longmapsto 2xe^x+2e^x`

`\\ \sf\longmapsto 2e^x(x+1)`