Differentiate

g(x)=x^(2^1/2)+(x^3/2^1/2)+e^(2^1/2)+(2e^x)^1/2

Let's rewrite that as

g(x) = x^(5/2) + x^(3/4) + sqrt2*e^(x/2) + e^5/2

Differentiate the terms one at a time and add the results. Since e^5/2 is a constant, the derivative of that term is zero. Remember the general rule for the derivative of x^n:
d/dx (x^n) = n x^(n-1)

I will be glad to critique your work

To differentiate the given function g(x), we will use the power rule for differentiation, as well as the chain rule and product rule when necessary.

The power rule states that if we have a term in the form of x^n, the derivative is given by nx^(n-1).

Let's break down the given function and differentiate each term separately:

1. Term 1: x^(2^1/2)
- Apply the power rule: The derivative of x^n is nx^(n-1).
- In this case, n is 2^1/2, so we differentiate with respect to x as follows: (2^1/2)x^(2^1/2 - 1).
- Simplifying, we get: (2^1/2)x^(1/2).

2. Term 2: (x^3/2^1/2)
- Here, we have a fraction as the exponent. To differentiate, we will use the power rule and the chain rule.
- First, apply the power rule: The derivative of x^n is nx^(n-1).
- In this case, n is 3/2, so we differentiate with respect to x as follows: (3/2)x^(3/2 - 1).
- Simplifying, we get: (3/2)x^(1/2).
- Now, we apply the chain rule. The chain rule states that if we have a term in the form of (f(g(x)))^n, the derivative is n(f(g(x)))^(n-1) * f'(g(x)) * g'(x).
- Our function is (x^3/2^1/2). Let f(u) = u^3/2 and g(x) = x^1/2.
- To apply the chain rule, we need to find f'(u) and g'(x).
- f'(u) = (3/2)u^(3/2 - 1) = (3/2)u^(1/2) = (3/2)sqrt(u).
- g'(x) = (1/2)x^(1/2 - 1) = (1/2)x^(-1/2) = (1/2)/sqrt(x).
- Now, we substitute f'(u) and g'(x) into the chain rule formula:
- (3/2)x^(1/2) * (1/2)/sqrt(x) = (3/4)sqrt(x).

3. Term 3: e^(2^1/2)
- The derivative of e^x is simply e^x. Therefore, the derivative of e^(2^1/2) is e^(2^1/2).

4. Term 4: (2e^x)^1/2
- Again, we have a fractional exponent. To differentiate, we can use the power rule and chain rule.
- First, apply the power rule: The derivative of x^n is nx^(n-1).
- In this case, n is 1/2, so we differentiate with respect to x as follows: (1/2)(2e^x)^(1/2 - 1).
- Simplifying, we get: (1/2)(2e^x)^(-1/2).
- Now, apply the chain rule. The function is (2e^x)^1/2. Let f(u) = u^(1/2) and g(x) = 2e^x.
- To apply the chain rule, we need to find f'(u) and g'(x).
- f'(u) = (1/2)u^(1/2 - 1) = (1/2)u^(-1/2) = (1/2)/sqrt(u).
- g'(x) = (2e^x)' = 2e^x.
- Substitute f'(u) and g'(x) into the chain rule formula:
- (1/2)/sqrt(2e^x) * 2e^x = e^x/sqrt(2e^x) = e^x/(sqrt(2)e^x) = 1/(sqrt(2)).

Now that we have differentiated each term, we can combine them to get the final derivative of g(x):

g'(x) = (2^1/2)x^(1/2) + (3/4)sqrt(x) + e^(2^1/2) + 1/(sqrt(2)).

Thus, the derivative of g(x) is (2^1/2)x^(1/2) + (3/4)sqrt(x) + e^(2^1/2) + 1/(sqrt(2)).