Problem 12.
Find the derivative of the function.
f(x)=e^(x)^1/2
My answer is: [e^(x)^1/2]/2x^(1/2)
Is this correct?
Yes, you are correct.
Thank you.
To find the derivative of the function f(x) = e^(x)^(1/2), we can use the chain rule. Let's go through the steps together.
Step 1: Start with the function f(x) = e^(x)^(1/2).
Step 2: Identify the composition of functions within the function. In this case, we have the outer function f(x) = e^x and the inner function g(x) = x^(1/2).
Step 3: Apply the chain rule, which states that if we have a composite function f(g(x)), then the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x).
First, let's find the derivative of the outer function f(x) = e^x. The derivative of e^x is simply e^x.
Next, let's find the derivative of the inner function g(x) = x^(1/2). We can rewrite it as g(x) = sqrt(x). The derivative of sqrt(x) with respect to x can be found using the power rule, which states that the derivative of x^n with respect to x is n * x^(n-1). Applying this rule, we have g'(x) = 1/2 * x^(-1/2).
Now, we can apply the chain rule by multiplying the derivatives of the outer and inner functions. f'(x) = e^x * (1/2 * x^(-1/2)).
So the correct answer is:
f'(x) = e^x * (1/2 * x^(-1/2)).
Note that your answer of [e^(x)^(1/2)] / (2 * x^(1/2)) is not correct. It seems like you applied the power rule incorrectly. Remember that the derivative of x^a is a * x^(a-1), not 1/a * x^(a-1).