Question

Find the derivatives w.r.t.x from first principle:
$\rm \sqrt{x+3}$

Answer 1

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Answer 2

Answer:

1/ 2√(x + 3)

Step-by-step explanation:

f(x)  = √(x + 3)

f'(x)  = Lim h - 0     (f(x + h) - f(x)) /h

=> f'(x)  = Lim h - 0   ( √(x + h + 3) - √(x + 3)) / h

Multiplying & dividing by √(x + h + 3) +  √(x + 3)

=> f'(x)  = Lim h - 0   ( √(x + h + 3) - √(x + 3)) / h   * (√(x + h + 3) +  √(x + 3))/(√(x + h + 3) +  √(x + 3))

=> f'(x)  = Lim h - 0  ((x + h + 3) - (x + 3))/( h (√(x + h + 3) +  √(x + 3)))

=> f'(x)  = Lim h - 0  h/( h (√(x + h + 3) +  √(x + 3)))

=> f'(x)  = Lim h - 0   1/ (√(x + h + 3) +  √(x + 3))

putting h = 0

=> f'(x)  = 1/ 2√(x + 3)