MHB At what point on the curve is there a tangent line parallel to the line

[tmt1]

At what point on the curve
$$y = e^x$$ is the tangent line parallel to the line

$$y = 2x$$

The derivative of y is

$$\frac{dy}{dx} = e^x$$

But I'm unsure how to proceed from here.

 

[Nono713]

At what point on the curve
$$y = e^x$$ is the tangent line parallel to the line

$$y = 2x$$

The derivative of y is

$$\frac{dy}{dx} = e^x$$

But I'm unsure how to proceed from here.

You need to equate the slope of the exponential function at a point $x$, which is $e^x$ (as you have found) with the slope of the line $y = 2x$, which is $2$. Hence $e^x = 2$, and solve for $x$. Then if you graph that, you will find that the tangent line to $e^x$ at that point $x$ is parallel to $y = 2x$ as they have the same slope :)