differentiate from first principle x^(5/2)

To differentiate \( x^{\frac{5}{2}} \) from first principles, we use the definition of the derivative:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

Here, \( f(x) = x^{\frac{5}{2}} \). Let's proceed step-by-step.

### Step 1: Set up the difference quotient
We need to calculate the difference quotient \( \frac{f(x+h) - f(x)}{h} \).

\[
f'(x) = \lim_{h \to 0} \frac{(x+h)^{\frac{5}{2}} - x^{\frac{5}{2}}}{h}
\]

### Step 2: Simplify the expression
The tricky part is simplifying \( (x+h)^{\frac{5}{2}} - x^{\frac{5}{2}} \). We'll approximate this difference using the binomial expansion for small \( h \). For \( (x+h)^{\frac{5}{2}} \), the binomial expansion gives:

\[
(x+h)^{\frac{5}{2}} = x^{\frac{5}{2}} + \frac{5}{2} x^{\frac{3}{2}} h + \text{higher order terms in } h
\]

So, we have:

\[
(x+h)^{\frac{5}{2}} - x^{\frac{5}{2}} \approx \frac{5}{2} x^{\frac{3}{2}} h
\]

### Step 3: Substitute and simplify
Substitute this approximation into the difference quotient:

\[
f'(x) = \lim_{h \to 0} \frac{\frac{5}{2} x^{\frac{3}{2}} h}{h}
\]

Cancel the \( h \) terms:

\[
f'(x) = \lim_{h \to 0} \frac{5}{2} x^{\frac{3}{2}}
\]

Since there are no more terms involving \( h \), we can remove the limit:

\[
f'(x) = \frac{5}{2} x^{\frac{3}{2}}
\]

### Final Answer:
\[
\frac{d}{dx} \left( x^{\frac{5}{2}} \right) = \frac{5}{2} x^{\frac{3}{2}}
\]

Would you like more details or have any questions about this solution?

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Here are 5 related questions:

1. How do you apply first principles to differentiate polynomials?
2. Can you use first principles to differentiate \( x^n \) for any real \( n \)?
3. How does the binomial expansion help in simplifying expressions during differentiation?
4. What is the general derivative formula for \( x^n \)?
5. How do higher-order derivatives work with fractional powers?

**Tip:** Always look for patterns when differentiating functions using first principles. It helps in recognizing standard forms and applying rules efficiently.

Learn how to differentiate x^(5/2) from first principles using the definition of the derivative and the binomial expansion. This detailed guide walks through key concepts like first principles of calculus and exponent rules, suitable for Grades 11-12 students.

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