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

Question

Ezekiel Alexander · Accepted Answer

#d/dx (2/(5x+1)^3) = - 10 /((5x+1)^4) #

By definition:

#f'(x) = lim_(h->0) (f(x+h)-f(x))/h#

so:

#d/dx (2/(5x+1)^3) = lim_(h->0) 2/h(1/(5(x+h)+1)^3-1/(5x+1)^3)#

regroup the binomials at the denominator

#d/dx (2/(5x+1)^3) = lim_(h->0) 2/h(1/((5x+1)+5h)^3-1/(5x+1)^3)#

Perform the difference:

#d/dx (2/(5x+1)^3) = lim_(h->0) 2/h( ( (5x+1)^3 - ((5x+1)+5h)^3 )/(((5x+1)+5h)^3(5x+1)^3))#

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

#d/dx (2/(5x+1)^3) = lim_(h->0) 2/h( ( (5x+1)^3 - (5x+1)^3 - 5h(5x+1)^2 - 25h^2(5x+1) -125h^3 )/(((5x+1)+5h)^3(5x+1)^3))#

The first two terms cancel each other:

#d/dx (2/(5x+1)^3) = lim_(h->0) -2/h( ( 5h(5x+1)^2 + 25h^2(5x+1) +125h^3 )/(((5x+1)+5h)^3(5x+1)^3))#

Simplifying #h# and separating the terms:

#d/dx (2/(5x+1)^3) = lim_(h->0) -( 10(5x+1)^2 )/(((5x+1)+5h)^3(5x+1)^3)-lim_(h->0) -( 50h(5x+1) )/(((5x+1)+5h)^3(5x+1)^3)-lim_(h->0) - ( 250h^2)/(((5x+1)+5h)^3(5x+1)^3)#

The last two limits are zero, so:

#d/dx (2/(5x+1)^3) = -( 10(5x+1)^2 )/((5x+1)^6) = - 10 /((5x+1)^4) #

Cameron Alexander · Answer

undefined 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} $.