Find the derivative using the chain rule

f(x) = 3sqrt(x^2-1)

To find the derivative of \( f(x) = 3\sqrt{x^2 - 1} \) using the chain rule, let's go step by step.

### Step 1: Rewrite the function
First, express the square root in exponential form:
\[
f(x) = 3(x^2 - 1)^{1/2}
\]

### Step 2: Apply the chain rule
The chain rule states that if you have a composition of functions, say \( f(g(x)) \), then the derivative is given by:
\[
f'(x) = f'(g(x)) \cdot g'(x)
\]

Here, \( f(x) = 3(u^{1/2}) \) and \( u = x^2 - 1 \). So, we'll differentiate the outer function first and then the inner function.

### Step 3: Differentiate the outer function
The outer function is \( 3u^{1/2} \), where \( u = x^2 - 1 \). The derivative of \( u^{1/2} \) with respect to \( u \) is:
\[
\frac{d}{du} \left( u^{1/2} \right) = \frac{1}{2} u^{-1/2}
\]
So the derivative of the outer function is:
\[
\frac{d}{dx} \left[ 3(x^2 - 1)^{1/2} \right] = 3 \cdot \frac{1}{2} (x^2 - 1)^{-1/2}
\]

### Step 4: Differentiate the inner function
Now, differentiate the inner function \( u = x^2 - 1 \):
\[
\frac{d}{dx}(x^2 - 1) = 2x
\]

### Step 5: Multiply them together
Using the chain rule, multiply the derivative of the outer function by the derivative of the inner function:
\[
f'(x) = 3 \cdot \frac{1}{2}(x^2 - 1)^{-1/2} \cdot 2x
\]

### Step 6: Simplify the expression
Now simplify the terms:
\[
f'(x) = 3x(x^2 - 1)^{-1/2}
\]

Thus, the derivative of \( f(x) = 3\sqrt{x^2 - 1} \) is:
\[
f'(x) = \frac{3x}{\sqrt{x^2 - 1}}
\]

### Final Answer:
\[
f'(x) = \frac{3x}{\sqrt{x^2 - 1}}
\]

Would you like further details on any part of this solution? 

Here are 5 related questions to explore:

1. How would you find the second derivative of \( f(x) = 3\sqrt{x^2 - 1} \)?
2. Can you apply the product rule to functions involving square roots and polynomials?
3. What happens if \( f(x) = \sqrt{x^2 + 1} \), and how does the derivative change?
4. How does the chain rule work when there are multiple layers of functions?
5. How does the chain rule apply in implicit differentiation?

**Tip:** When dealing with square roots, rewriting them as exponents (like \( u^{1/2} \)) often simplifies the process of taking derivatives.

Find the derivative using the chain rule
f(x) = 3sqrt(x^2-1)

Learn how to find the derivative of f(x) = 3sqrt(x^2 - 1) using the chain rule with a detailed step-by-step solution. This problem involves key calculus concepts such as the chain rule and simplifying square roots. Suitable for high school or early college calculus students.

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