Given the function f(x) = 3/sqrt(x^3) find f'(x)

To find the derivative of the function \( f(x) = \frac{3}{\sqrt{x^3}} \), let's first rewrite the expression in a form that is easier to differentiate.

### Step 1: Rewrite the function
We can rewrite \( \frac{3}{\sqrt{x^3}} \) as a power of \( x \):
\[
f(x) = 3x^{-\frac{3}{2}}
\]
since \( \sqrt{x^3} = x^{\frac{3}{2}} \) and dividing by \( x^{\frac{3}{2}} \) is the same as multiplying by \( x^{-\frac{3}{2}} \).

### Step 2: Differentiate using the power rule
The power rule for differentiation states that if \( f(x) = x^n \), then \( f'(x) = n x^{n-1} \).

Now apply the power rule to \( f(x) = 3x^{-\frac{3}{2}} \):

\[
f'(x) = 3 \cdot \left( -\frac{3}{2} \right) x^{-\frac{3}{2} - 1}
\]
\[
f'(x) = -\frac{9}{2} x^{-\frac{5}{2}}
\]

### Step 3: Simplify the result
So, the derivative of the function is:
\[
f'(x) = -\frac{9}{2x^{\frac{5}{2}}}
\]

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. What is the second derivative of \( f(x) = \frac{3}{\sqrt{x^3}} \)?
2. How do you interpret the derivative of this function in terms of rate of change?
3. What happens to \( f'(x) \) as \( x \to 0 \) and \( x \to \infty \)?
4. How would you compute the derivative of \( f(x) = \frac{3}{x^3} \)?
5. How can we apply this concept to find the tangent line to \( f(x) \) at a specific point?

**Tip**: When dealing with square roots in functions, it can be helpful to rewrite them as fractional exponents to simplify differentiation.

Learn how to find the derivative of the function f(x) = 3/sqrt(x^3) with a detailed, step-by-step solution. This problem involves calculus concepts, such as differentiation and the power rule, making it ideal for high school students in Grades 11-12.

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