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.