Question

The equation of the tangent to the curve $y=x{e}^x$ at x = 0 is .

• A. y = x
• B. y = 2x
• C. x + y = 0

Answer 1

The correct option is A y = x

The given curve is $y=x{e}^x$

When x = 0, $y=0\times e^0=0$

Also, the sleope of the tangent at any point $(x_0,y_0)$ is given by

$slope=\frac{dy}{dx}{|}_{(x_0,y_0)}$

Here, $\frac{dy}{dx}=\frac{d\left(x{e}^x\right)}{dx}$

$=x\times \frac{d}{dx}\left(e^x\right)+e^x\times \frac{d}{dx}\left(x\right)$

$=x{e}^x+e^x$
$=e^x(1+x)$

Thus, at (0,0), $\frac{dy}{dx}=e^0(1+0)=1$

The slope of the tangent at (0,0) is 1

Thus, the equation of the tangent is $y-0=1(x-0)$
Which is y = x.