use first principles to find f'(x) when f(x) is 1/x

To find the derivative of \( f(x) = \frac{1}{x} \) using first principles (i.e., the definition of the derivative), we use the formula:

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

### Step 1: Write \( f(x) \) and \( f(x+h) \)
Given that \( f(x) = \frac{1}{x} \), we calculate \( f(x+h) \):
\[
f(x+h) = \frac{1}{x+h}
\]

### Step 2: Apply the formula
Substitute \( f(x) = \frac{1}{x} \) and \( f(x+h) = \frac{1}{x+h} \) into the definition of the derivative:
\[
f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h}
\]

### Step 3: Simplify the expression
Combine the two fractions in the numerator:
\[
f'(x) = \lim_{h \to 0} \frac{\frac{x - (x+h)}{(x+h)x}}{h}
= \lim_{h \to 0} \frac{\frac{-h}{(x+h)x}}{h}
\]

### Step 4: Simplify further
\[
f'(x) = \lim_{h \to 0} \frac{-h}{h(x+h)x}
\]

Cancel the \( h \)'s in the numerator and denominator:
\[
f'(x) = \lim_{h \to 0} \frac{-1}{(x+h)x}
\]

### Step 5: Take the limit as \( h \to 0 \)
As \( h \) approaches 0, \( x+h \) approaches \( x \), so the expression simplifies to:
\[
f'(x) = \frac{-1}{x^2}
\]

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

Would you like any further details or clarifications?

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Here are 5 related questions for further practice:
1. Use first principles to differentiate \( f(x) = x^2 \).
2. Find the derivative of \( f(x) = \frac{1}{x^2} \) using first principles.
3. What is the second derivative of \( f(x) = \frac{1}{x} \)?
4. Differentiate \( f(x) = \ln(x) \) using the definition of a derivative.
5. Use first principles to find the derivative of \( f(x) = \frac{1}{x+1} \).

**Tip:** Mastering first principles helps in understanding how all differentiation rules are derived, enhancing your problem-solving flexibility.

Use first principles to find f'(x) when f(x) is 1/x

Discover how to use first principles to differentiate f(x) = 1/x. This step-by-step solution explores key calculus concepts like limits and derivatives. Suitable for students in Grades 11-12 looking to deepen their understanding of foundational calculus.

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