How do you use the limit definition of the derivative to find the derivative of #f(x)=sqrt(4x-5)#?

Question

Ezekiel Alexander · Accepted Answer

#d/dx( sqrt(4x-5)) = 2/ sqrt(4x-5)#

By definition of derivative we have:

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

so:

#d/dx( sqrt(4x-5)) = lim_(h->0) (sqrt(4(x+h)-5)-sqrt(4x-5))/h#

rationalize the numerator using the identity:

#a^2-b^2 =(a-b)(a+b)#

#d/dx( sqrt(4x-5)) = lim_(h->0) ((sqrt(4(x+h)-5)-sqrt(4x-5))/h)( (sqrt(4(x+h)-5)+sqrt(4x-5))/(sqrt(4(x+h)-5)+sqrt(4x-5)))#

#d/dx( sqrt(4x-5)) = lim_(h->0) (4(x+h)-5-(4x-5))/(h (sqrt(4(x+h)-5)+sqrt(4x-5))#

#d/dx( sqrt(4x-5)) = lim_(h->0) (cancel(4x)+4h-cancel5-cancel(4x)+cancel5)/(h (sqrt(4(x+h)-5)+sqrt(4x-5))#

#d/dx( sqrt(4x-5)) = lim_(h->0) (4cancelh)/(cancelh (sqrt(4(x+h)-5)+sqrt(4x-5))#

#d/dx( sqrt(4x-5)) = lim_(h->0) 4/ (sqrt(4(x+h)-5)+sqrt(4x-5)#

#d/dx( sqrt(4x-5)) = 2/ sqrt(4x-5)#

Ezra Adams · Answer

undefined To use the limit definition of the derivative to find the derivative of f(x) = √(4x - 5), follow these steps:

1. Start with the limit definition of the derivative: f'(x) = lim(h->0) [f(x + h) - f(x)] / h.

2. Substitute f(x) into the formula: f'(x) = lim(h->0) [√(4(x + h) - 5) - √(4x - 5)] / h.

3. Simplify the expression by expanding the square roots: f'(x) = lim(h->0) [(√(4x + 4h - 5) - √(4x - 5))] / h.

4. Rationalize the numerator by multiplying by the conjugate: f'(x) = lim(h->0) [(√(4x + 4h - 5) - √(4x - 5)) * (√(4x + 4h - 5) + √(4x - 5))] / (h * (√(4x + 4h - 5) + √(4x - 5))).

5. Expand and simplify the expression: f'(x) = lim(h->0) [(4x + 4h - 5) - (4x - 5)] / (h * (√(4x + 4h - 5) + √(4x - 5))).

6. Cancel out like terms: f'(x) = lim(h->0) [4h] / (h * (√(4x + 4h - 5) + √(4x - 5))).

7. Simplify further: f'(x) = lim(h->0) [4] / (√(4x + 4h - 5) + √(4x - 5)).

8. Plug in h = 0: f'(x) = 4 / (√(4x - 5) + √(4x - 5)).

9. Simplify the expression: f'(x) = 4 / (2√(4x - 5)).

10. Finally, simplify further: f'(x) = 2 / √(4x - 5).