Home > Calculus > Differentiating Exponential Functions with Base e

How do you find the derivative of # y=e^(x^(1/2))#?

Answer 1

Samantha Adkison

#e^sqrt(x)/(2sqrt(x))#

A substitution here would help tremendously!

Let's say that #x^(1/2) = u#

now,

#y = e^u#

We know that the derivative of #e^x# is #e^x# so;

#dy/dx = e^u * (du)/dx# using the chain rule

#d/dx x^(1/2) = (du)/dx = 1/2*x^(-1/2) = 1/(2sqrt(x))#

Now plug #(du)/dx# and #u# back into the equation :D

#dy/dx = e^sqrt(x)/(2sqrt(x))#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Aurora Alvarado

To find the derivative of ( y = e^{x^{1/2}} ), use the chain rule:

( \frac{dy}{dx} = \frac{d}{dx}(e^{x^{1/2}}) = e^{x^{1/2}} \cdot \frac{d}{dx}(x^{1/2}) )

Apply the power rule to ( x^{1/2} ):

( \frac{dy}{dx} = e^{x^{1/2}} \cdot \frac{1}{2}x^{-1/2} )

So, the derivative is:

( \frac{dy}{dx} = \frac{e^{x^{1/2}}}{2\sqrt{x}} )

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.