How do you use the limit definition to find the derivative of #f(x)=2/(5x+1)^3#?

By definition:

so:

regroup the binomials at the denominator

Perform the difference:

Expand now the power of the second binomial at the numerator:

The first two terms cancel each other:

The last two limits are zero, so:

To find the derivative of $f(x) = \frac{2}{(5x + 1)^3}$ using the limit definition:

1. Start with the definition of the derivative:
   $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

2. Substitute $f(x) = \frac{2}{(5x + 1)^3}$ into the formula:
   $f'(x) = \lim_{h \to 0} \frac{\frac{2}{(5(x+h) + 1)^3} - \frac{2}{(5x + 1)^3}}{h}$

3. Find a common denominator for the fractions:
   $f'(x) = \lim_{h \to 0} \frac{2((5x + 1)^3 - (5(x+h) + 1)^3)}{h((5(x + h) + 1)^3 \cdot (5x + 1)^3)}$

4. Expand the binomials:
   $f'(x) = \lim_{h \to 0} \frac{2(125x^3 + 75x^2 + 15x + 1 - (125(x^3 + 3x^2h + 3xh^2 + h^3) + 75(x^2 + 2xh + h^2) + 15(x + h) + 1))}{h((5(x + h) + 1)^3 \cdot (5x + 1)^3)}$

5. Simplify the expression:
   $f'(x) = \lim_{h \to 0} \frac{2(125x^3 + 75x^2 + 15x + 1 - (125x^3 + 375x^2h + 375xh^2 + 125h^3 + 75x^2 + 150xh + 75h^2 + 15x + 15h + 1))}{h((5(x + h) + 1)^3 \cdot (5x + 1)^3)}$

$f'(x) = \lim_{h \to 0} \frac{2(75x^2 - 375x^2h + 15x - 150xh + 15h - 125h^2)}{h((5(x + h) + 1)^3 \cdot (5x + 1)^3)}$

6. Cancel out common terms:
   $f'(x) = \lim_{h \to 0} \frac{2(-375x^2h - 150xh - 125h^2)}{h((5(x + h) + 1)^3 \cdot (5x + 1)^3)}$

$f'(x) = \lim_{h \to 0} \frac{-750x^2 - 300xh - 250h^2}{(5(x + h) + 1)^3 \cdot (5x + 1)^3}$

7. Apply the limit:
   $f'(x) = \frac{-750x^2}{(5x + 1)^3 \cdot (5x + 1)^3}$

$f'(x) = \frac{-750x^2}{(5x + 1)^6}$

Therefore, the derivative of $f(x) = \frac{2}{(5x + 1)^3}$ is $f'(x) = \frac{-750x^2}{(5x + 1)^6}$.