Find the value of (f o g)' at x = 4 when f(u)= cos(piu/24) and u=g(x)=6sqrt(x)
y = cos (6 pi x^.5/24) = cos ( pi x^.5/4)
let z = x^.5 then dz/dx = .5 x^-.5
y = cos( pi z/4)
dy/dz = -(pi/4)sin (pi z/4)
dy/dx = dy/dz * dz/dx
dy/dx = [-(pi/4)sin (pi x^.5 /4)] .5 x^-.5
if x = 4 then x^.5 = 2
dy/dx = [-pi/4 sin (pi/2) ] (1/4) = -pi/16
Well, let's start with some clown math!
To find (f o g)' at x = 4, we need to find the derivative of f(g(x)), and then evaluate it at x = 4.
First, let's find the derivative of f(u):
f'(u) = -sin(piu/24) * (pi/24)
Now, we need to find the derivative of g(x):
g'(x) = 6 / (2 * sqrt(x))
Finally, let's combine these derivatives to find (f o g)':
(f o g)'(x) = f'(g(x)) * g'(x)
Substituting the derivatives we found earlier:
(f o g)'(x) = [-sin(pi * (6sqrt(x))/24) * (pi/24)] * [6 / (2 * sqrt(x))]
Now, let's evaluate this at x = 4:
(f o g)'(4) = [-sin(pi * (6sqrt(4))/24) * (pi/24)] * [6 / (2 * sqrt(4))]
Simplifying this clown expression, we get:
(f o g)'(4) = [-sin(3pi/24) * (pi/24)] * [6 / (2 * 2)]
Now, let's go even more clownish and simplify further:
(f o g)'(4) = [-sin(pi/8) * (pi/24)] * [6/4]
Finally, let's get the ultimate clown answer:
(f o g)'(4) = (pi/24) * (-sin(pi/8)) * (3/2)
So, the value of (f o g)' at x = 4 is (pi/24) * (-sin(pi/8)) * (3/2).
To find the value of (f o g)' at x = 4, we need to calculate the derivative of f(g(x)) and then substitute x = 4 into the derivative.
Step 1: Find the derivative of f(u)
Let's find the derivative of f(u) = cos(piu/24) with respect to u.
Using the chain rule, we have:
(f'(u)) = -(pi/24)sin(piu/24)
Step 2: Find the derivative of g(x)
Given g(x) = 6√(x), we need to find g'(x), the derivative of g(x) with respect to x.
Using the power rule and the chain rule, we have:
(g'(x)) = 3/√(x)
Step 3: Calculate (f o g)'(x)
To find (f o g)'(x), we substitute g(x) into f'(u) and then multiply by g'(x):
(f o g)'(x) = (f'(u)) * (g'(x))
= (-(pi/24)sin(piu/24)) * (3/√(x))
Step 4: Substitute x = 4
Now, substitute x = 4 into (f o g)'(x):
(f o g)'(4) = (-(pi/24)sin(pi(6√4)/24)) * (3/√(4))
Step 5: Simplify the expression (if needed)
Let's simplify this expression further:
(f o g)'(4) = (-(pi/24)sin(pi(6√4)/24)) * (3/√(4))
= (-(pi/24)sin(pi(6*2)/24)) * (3/2)
= (-(pi/24)sin(12pi/24)) * (3/2)
= (-(pi/24)sin(1/2 * pi)) * (3/2)
= (-pi/24) * (1) * (3/2)
= -pi/16
Therefore, the value of (f o g)' at x = 4 is -pi/16.
To find the value of (f o g)' at x = 4, we need to find the derivative of the composite function f(g(x)) and then substitute x = 4.
Step 1: Find the derivative of f(u) with respect to u.
The given function f(u) = cos(piu/24). To find its derivative, we can use the chain rule.
The chain rule states that if we have a composite function y = f(g(x)), then the derivative dy/dx is given by dy/dx = (dy/du) * (du/dx).
Applying the chain rule to f(u) = cos(piu/24), we get:
df/du = (-sin(piu/24)) * (pi/24) [derivative of cos(u) = -sin(u), derivative of u = pi/24]
So, df/du = -(pi/24) * sin(piu/24).
Step 2: Find the derivative of g(x) with respect to x.
The given function g(x) = 6sqrt(x). To find its derivative, we can use the power rule and the chain rule.
Differentiating, we get:
dg/dx = 6 * (1/2) * x^(-1/2) = 3 / sqrt(x).
Step 3: Substitute x = 4 and find f'(g(x)).
Substituting x = 4, we get:
g(4) = 6sqrt(4) = 6 * 2 = 12.
Now, substitute u = g(x) = 12 in the derivative we found in Step 1:
df/du = -(pi/24) * sin(pi * u / 24) = -(pi/24) * sin(pi * 12 / 24) = -(pi/24) * sin(pi/2) = -(pi/24).
Therefore, (f o g)' at x = 4 is equal to df/du evaluated at u = 12, which we found to be -(pi/24).
Hence, the value of (f o g)' at x = 4 is -(pi/24).