f'(x) of ( x(sin(x^3))^3)/ ((x^7) + 1)^1/2

To find the derivative \( f'(x) \) of the function 

\[
f(x) = \frac{x (\sin(x^3))^3}{\sqrt{x^7 + 1}},
\]

we'll need to apply a combination of product, chain, and quotient rules. Let's break it down step-by-step.

### Step 1: Identify the quotient
We can treat the function as a quotient of two parts:

\[
f(x) = \frac{u(x)}{v(x)},
\]

where

\[
u(x) = x (\sin(x^3))^3
\]
and
\[
v(x) = \sqrt{x^7 + 1} = (x^7 + 1)^{1/2}.
\]

### Step 2: Derivative using the Quotient Rule
The quotient rule is:

\[
\frac{d}{dx} \left(\frac{u(x)}{v(x)}\right) = \frac{v(x) u'(x) - u(x) v'(x)}{v(x)^2}.
\]

We need to calculate \( u'(x) \) and \( v'(x) \) next.

### Step 3: Differentiate \( u(x) = x (\sin(x^3))^3 \)
Using the product rule for derivatives:

\[
u'(x) = \frac{d}{dx} \left( x (\sin(x^3))^3 \right) = \left( \frac{d}{dx} (x) \right) (\sin(x^3))^3 + x \frac{d}{dx} \left( (\sin(x^3))^3 \right).
\]

- The derivative of \( x \) is 1, so the first term is:

\[
(\sin(x^3))^3.
\]

- For the second term, use the chain rule:

\[
\frac{d}{dx} \left( (\sin(x^3))^3 \right) = 3(\sin(x^3))^2 \cdot \frac{d}{dx} \left( \sin(x^3) \right).
\]

- The derivative of \( \sin(x^3) \) is \( \cos(x^3) \cdot 3x^2 \) by the chain rule. So, the second term becomes:

\[
3(\sin(x^3))^2 \cdot \cos(x^3) \cdot 3x^2 = 9x^2 (\sin(x^3))^2 \cos(x^3).
\]

Therefore,

\[
u'(x) = (\sin(x^3))^3 + 9x^3 (\sin(x^3))^2 \cos(x^3).
\]

### Step 4: Differentiate \( v(x) = (x^7 + 1)^{1/2} \)
Using the chain rule:

\[
v'(x) = \frac{1}{2} (x^7 + 1)^{-1/2} \cdot 7x^6 = \frac{7x^6}{2\sqrt{x^7 + 1}}.
\]

### Step 5: Apply the Quotient Rule
Now that we have \( u'(x) \) and \( v'(x) \), we can plug them into the quotient rule:

\[
f'(x) = \frac{\sqrt{x^7 + 1} \cdot \left[ (\sin(x^3))^3 + 9x^3 (\sin(x^3))^2 \cos(x^3) \right] - x (\sin(x^3))^3 \cdot \frac{7x^6}{2\sqrt{x^7 + 1}}}{(x^7 + 1)}.
\]

### Final Answer:
\[
f'(x) = \frac{\sqrt{x^7 + 1} \cdot \left[ (\sin(x^3))^3 + 9x^3 (\sin(x^3))^2 \cos(x^3) \right] - \frac{7x^7 (\sin(x^3))^3}{2\sqrt{x^7 + 1}}}{x^7 + 1}.
\]

Would you like more details on any part of the process, or help with simplifications?

### 5 Related Questions:
1. How do you apply the chain rule in differentiating composite functions?
2. What is the general form of the quotient rule in calculus?
3. Can you differentiate \( \sin(x^3) \) using the chain rule?
4. What are common mistakes when using the product rule for derivatives?
5. How do you simplify expressions involving complex trigonometric derivatives?

### Tip:
Always group similar terms when applying the product and quotient rules to avoid confusion in long derivative expressions!

Learn how to compute the derivative of the function (x(sin(x^3))^3) / sqrt(x^7 + 1) using the quotient rule, chain rule, and product rule. This step-by-step solution explores key calculus concepts and is ideal for college-level students in Calculus I or II.

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