let y=(x+sin(x))^2
f(x)=?
g(x)=?
(f o g)'=?
compute the derivative using the chain rule
g(x) = x + sin x
f(x) = x^2
y = f(g(x)) = 2(x + sin x)(1 + cos x)
Well, well, well, look who's getting all mathematical! Let's break it down step by step, shall we?
First, let's find f(x). We have f(x) = (x + sin(x))^2.
Now, let's move onto g(x). Since g(x) is not given, we can choose any function we like, let's say g(x) = x.
Now, using the chain rule, we can find (f o g)'. The chain rule states that if h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).
To apply the chain rule, we need to differentiate f(x) with respect to g(x), and then multiply it by g'(x).
Let's start with finding f'(x). Using the power rule, we have f'(x) = 2(x + sin(x)) * (1 + cos(x)).
Since g(x) = x, we have g'(x) = 1.
Now, we can apply the chain rule: (f o g)' = f'(g(x)) * g'(x) = 2(g(x) + sin(g(x))) * (1 + cos(g(x))) * g'(x).
Substituting g(x) = x, we get (f o g)' = 2(x + sin(x)) * (1 + cos(x)) * 1.
So, the derivative of (f o g) is 2(x + sin(x)) * (1 + cos(x)).
Voila! Derivatives, chain rule, and a touch of comedy. Hope this helps!
To compute the derivatives of the given functions, let's go step-by-step:
1. Function y = (x + sin(x))^2:
To compute f(x), we just need to rewrite the given expression, y, without using the variable y. Therefore:
f(x) = (x + sin(x))^2
2. Function g(x):
The function g(x) is not provided in the question. Could you please provide the expression or value of g(x)?
3. (f o g)': Derivative using the chain rule:
To calculate the derivative of (f o g), we need to apply the chain rule.
Let's assume that g(x) is a differentiable function and f(x) is a differentiable function of g.
The chain rule states that if y = f(g(x)), then the derivative of y with respect to x is given by:
dy/dx = dy/dg * dg/dx
In this case, we can find (f o g)':
(f o g)' = df/dx = df/dg * dg/dx
Here, we are given g(x), so we need to find df/dg and dg/dx separately.
df/dg:
To find df/dg, we need to differentiate f(x) with respect to g and multiply it with the derivative of g(x) (dg/dx). Since we have f(x) = (x + sin(x))^2, and g(x) is assumed to be a function of x, we can differentiate f(x) with respect to g as if g were a constant.
df/dg = 2(x + sin(x))
dg/dx:
We are still awaiting the expression for the function g(x) to find its derivative (dg/dx). Once we have that expression, we can calculate it.
Once we have df/dg and dg/dx, we can substitute these values back into the chain rule formula to find the derivative (f o g)'.
To find the derivative of the given functions and their composition, we'll need to apply the chain rule. Let's start by finding f(x) and g(x) first.
Given:
f(x) = (x + sin(x))^2
g(x) = x
To find f(x), we substitute (x + sin(x))^2 for f(x):
f(x) = ((x + sin(x))^2)^2
= (x + sin(x))^4
To find g(x), we already have it:
g(x) = x
Now, let's find (f o g)':
(f o g)' represents the derivative of the composition of functions f and g.
(f o g)' = f'(g(x)) * g'(x)
To compute (f o g)', we need to find the derivatives f'(x) and g'(x):
f'(x) = 4(x + sin(x))^3 * (1 + cos(x))
g'(x) = 1
Now, we substitute these derivatives into the chain rule formula:
(f o g)' = f'(g(x)) * g'(x)
= f'(g(x)) * 1
= f'(g(x))
Since g(x) = x, we have:
(f o g)' = f'(x)
Therefore, (f o g)' is equal to the derivative of f(x), which is f'(x):
(f o g)' = f'(x)
= 4(x + sin(x))^3 * (1 + cos(x))
Hence, the derivative of (f o g) using the chain rule is 4(x + sin(x))^3 * (1 + cos(x)).