Given the following functions: F(u)= u^(5/2) and g(x)= x^6+1, find-f(g(x))=?, f'(u)=?, f'(g(x))=?, g'(x)=?,(f o g)'(x)=?
f(g) = (fog) = g^(5/2) = (x^6 + 1)^(5/2)
f'(u) = /2 u^(3/2)
f'(g(x)) = 5/2 g^(3/2) g' = 5/2 (x^6+1)^(3/2) * 6x^5
g'(x) = 6x^5
(fog)' = 5/2 (x^6+1)^(3/2) * 6x^5
Oh boy, it's time for some function shenanigans! Let's dive right in!
First, let's tackle -f(g(x)). To find this, we insert g(x) into f(u) and then negate the result. So, -f(g(x)) equals -(g(x))^(5/2).
Next up, let's find f'(u). The derivative of F(u) = u^(5/2) is f'(u) = (5/2)u^(3/2). Not too complicated, right?
Now, let's determine f'(g(x)). We need to take the derivative of f(u) with respect to u and then substitute g(x) in place of u. So, f'(g(x)) = (5/2)(g(x))^(3/2).
Moving on, let's tackle g'(x). To find this, we differentiate g(x) = x^6 + 1. The derivative of x^6 is 6x^5, and the derivative of 1 is 0. Therefore, g'(x) = 6x^5.
Lastly, let's figure out (f o g)'(x), which represents the derivative of the composition of f(g(x)). We can simplify this as (f o g)'(x) = f'(g(x)) * g'(x). So, if we substitute the values we found earlier, we get (f o g)'(x) = (5/2)(g(x))^(3/2) * 6x^5.
And there you have it! We've clowned our way through these functions, navigating derivatives and compositions. Hope this answered your question with a smile!
To find f(g(x)), we substitute g(x) into f(u) as follows:
f(g(x)) = (g(x))^(5/2) = (x^6 + 1)^(5/2).
To calculate f'(u), we differentiate the function F(u) with respect to u:
f'(u) = d/dx (u^(5/2)).
Using the power rule for differentiation, we have:
f'(u) = (5/2) * u^(5/2 - 1) = (5/2) * u^(3/2).
To compute f'(g(x)), we need to apply the chain rule. That is:
f'(g(x)) = f'(u) * g'(x).
To calculate g'(x), we differentiate the function g(x) with respect to x:
g'(x) = d/dx (x^6 + 1).
Using the power rule for differentiation, we get:
g'(x) = 6 * x^(6 - 1) = 6 * x^5.
Finally, to find (f o g)'(x), which represents the derivative of f(g(x)), we substitute f'(g(x)) and g'(x) into the chain rule formula:
(f o g)'(x) = f'(g(x)) * g'(x).
So, (f o g)'(x) = f'(g(x)) * g'(x) = (5/2) * (x^6 + 1)^(3/2) * 6 * x^5.
Therefore, the answers are:
f(g(x)) = (x^6 + 1)^(5/2),
f'(u) = (5/2) * u^(3/2),
f'(g(x)) = (5/2) * (x^6 + 1)^(3/2),
g'(x) = 6 * x^5,
(f o g)'(x) = (5/2) * (x^6 + 1)^(3/2) * 6 * x^5.
To find the requested values, we will use some basic calculus rules.
1. Find f(g(x)):
To find f(g(x)), we need to substitute g(x) into f(u):
f(g(x)) = (g(x))^(5/2)
= (x^6 + 1)^(5/2)
2. Find f'(u):
To find f'(u), we differentiate the function f(u):
f'(u) = d(u^(5/2))/du
= (5/2)u^(5/2 - 1)
= (5/2)u^(3/2)
3. Find f'(g(x)):
To find f'(g(x)), we need to differentiate f(u) and substitute g(x) into the derivative:
f'(g(x)) = (5/2)(g(x))^(3/2)
= (5/2)(x^6 + 1)^(3/2)
4. Find g'(x):
To find g'(x), we differentiate the function g(x):
g'(x) = d(x^6 + 1)/dx
= 6x^5
5. Find (f o g)'(x):
To find (f o g)'(x), we need to find the composition of f'(u) and g'(x):
(f o g)'(x) = f'(g(x)) * g'(x)
= (5/2)(x^6 + 1)^(3/2) * 6x^5
Therefore:
- f(g(x)) = (x^6 + 1)^(5/2)
- f'(u) = (5/2)u^(3/2)
- f'(g(x)) = (5/2)(x^6 + 1)^(3/2)
- g'(x) = 6x^5
- (f o g)'(x) = (5/2)(x^6 + 1)^(3/2) * 6x^5