Using first principles, differentiate. Show your work. [Hint use the ‘difference of cubes’]

To differentiate a function using first principles, we start by using the definition of the derivative. Given a function \( f(x) \), the derivative \( f'(x) \) at a point \( x \) is defined as:

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

Let’s differentiate \( f(x) = x^3 \) using first principles. The function is:

\[
f(x) = x^3
\]

### Step 1: Compute \( f(x+h) \)

First, substitute \( x+h \) into the function:

\[
f(x+h) = (x+h)^3
\]

### Step 2: Expand \( (x+h)^3 \)

Using the binomial theorem or the formula for the expansion of a cube, we have:

\[
(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3
\]

### Step 3: Apply the Difference Quotient

Substitute \( f(x+h) \) and \( f(x) \) into the difference quotient:

\[
\frac{f(x+h) - f(x)}{h} = \frac{(x^3 + 3x^2h + 3xh^2 + h^3) - x^3}{h}
\]

Simplify the expression by canceling out \( x^3 \):

\[
= \frac{3x^2h + 3xh^2 + h^3}{h}
\]

### Step 4: Factor Out \( h \)

Factor \( h \) from the numerator:

\[
= \frac{h(3x^2 + 3xh + h^2)}{h}
\]

Cancel the \( h \) from the numerator and denominator:

\[
= 3x^2 + 3xh + h^2
\]

### Step 5: Take the Limit as \( h \to 0 \)

Finally, take the limit of the expression as \( h \) approaches 0:

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

When \( h \) approaches 0, the terms involving \( h \) vanish:

\[
f'(x) = 3x^2
\]

Thus, the derivative of \( f(x) = x^3 \) using first principles is \( f'(x) = 3x^2 \).

### Summary:

The steps above demonstrate the use of the difference of cubes and the first principles to differentiate \( x^3 \). The final result is:

\[
f'(x) = 3x^2
\]

Would you like any further details, or do you have any questions?

### Relative Questions:
1. How would you differentiate \( x^4 \) using first principles?
2. Can you apply the first principles method to differentiate \( \sin(x) \)?
3. What would happen if we differentiate \( x^3 \) using the power rule instead of first principles?
4. How does the first principle method apply to a linear function like \( f(x) = 2x + 1 \)?
5. Can the first principles method be used to differentiate a piecewise function?

### Tip:
Always start with expanding and simplifying the expression before applying the limit when using first principles for differentiation.

Learn how to differentiate the function x^3 using first principles. This detailed guide covers the difference of cubes method, the definition of derivative, and the difference quotient formula. Suitable for advanced high school students and anyone interested in calculus fundamentals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation