To differentiate the function $f(x) = \frac{3x^3}{4}$ using first principles, we begin with the definition of the derivative:

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

Given the function $f(x) = \frac{3x^3}{4}$, we substitute into the definition:

$f'(x) = \lim_{h \to 0} \frac{\frac{3(x+h)^3}{4} - \frac{3x^3}{4}}{h}$

First, let's expand $(x+h)^3$ using the binomial theorem:

$(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$

Substitute this into the expression for the derivative:

$f'(x) = \lim_{h \to 0} \frac{\frac{3(x^3 + 3x^2h + 3xh^2 + h^3)}{4} - \frac{3x^3}{4}}{h}$

Simplify the expression:

$f'(x) = \lim_{h \to 0} \frac{\frac{3x^3 + 9x^2h + 9xh^2 + 3h^3}{4} - \frac{3x^3}{4}}{h}$

$f'(x) = \lim_{h \to 0} \frac{\frac{3x^3 + 9x^2h + 9xh^2 + 3h^3 - 3x^3}{4}}{h}$

$f'(x) = \lim_{h \to 0} \frac{\frac{9x^2h + 9xh^2 + 3h^3}{4}}{h}$

Now, factor $h$ out from the numerator:

$f'(x) = \lim_{h \to 0} \frac{h\left(\frac{9x^2 + 9xh + 3h^2}{4}\right)}{h}$

Since $h$ is in both the numerator and the denominator, we can cancel it out:

$f'(x) = \lim_{h \to 0} \frac{9x^2 + 9xh + 3h^2}{4}$

Now, take the limit as $h$ approaches 0:

$f'(x) = \frac{9x^2 + 9x(0) + 3(0)^2}{4} = \frac{9x^2}{4}$

Therefore, the derivative of the function $f(x) = \frac{3x^3}{4}$ is:

$f'(x) = \frac{9x^2}{4}$

Summary:

• We expanded $(x+h)^3$ using the binomial theorem.
• Simplified the expression using algebraic manipulation.
• Applied the limit as $h$ approaches 0 to obtain the final derivative.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How would you differentiate $g(x) = x^3$ using first principles?
2. What is the derivative of $f(x) = \frac{5x^4}{2}$ using first principles?
3. Can you explain the difference of cubes formula and how it applies here?
4. What is the general rule for differentiating a cubic polynomial?
5. How does using first principles differ from using standard differentiation rules?

Tip: Understanding first principles is fundamental as it forms the basis of all differentiation techniques.