Find the slope of the curve y=e^4x at x=0

Answer 1

To find the slope of the curve y = e^4x at x=0, we can first take the derivative of the function since taking the derivative of any function gives us the slope of the tangent line at any specific point on a curve. So, we draw a tangent line that goes through x=0 on that curve and we are basically finding the slope of that point.

We can find the derivative by applying the chain rule here which is used when you are differentiating a function inside another function (composite function). In this case 4x is inside of e^x.

Chain rule:

If y = f(g(x))

then dy/dx = f'(g(x)) • g'(x)

Lets apply chain rule:

Inside: g(x) = 4x → g'(x) = 4

Outside: f(u) = e^u → f'(u) = e^u

dy/dx = e^4x • 4 = 4e^4x

Can also be written as:

y' = 4e^4x

Now, since we are finding the slope at x=0, we can plug in 0 for x into the derivative:

y'(0) = 4e⁴⁽⁰⁾ = 4e⁰ = 4(1) = 4

So, the slope of the curve y = e^4x at x = 0 is 4.

Answer 2

If the equation is y = e^4x=(x)(e^4) then

dy/dx = d/dx(e^4(x)) = e^4 => the slope = e^4 for any value of x. It is a horizontal line of y = e^4

But if the equation is y = e^(4x) then

dy/dx = 4e^(4x). slope when x = 0 is 4(e^0) = 4 . It is still horizontal line at y = 4