Given f(x) = (1)/(sqrt(x+3)), we have f(x+h) = (1)/(sqrt(x+h+3)).

Step 1: Write the definition of the derivative from first principles. The derivative of a function f(x) is given by: f'(x) = _h 0 (f(x+h) - f(x))/(h) Given f(x) = (1)/(sqrt(x+3)), we have f(x+h) = (1)/(sqrt(x+h+3)). Substitute these into the formula: f'(x) = _h 0 (1)/(sqrt(x+h+3)) - (1)/(sqrt(x+3))h Step 2: Combine the fractions in the numerator. f'(x) = _h 0 sqrt(x+3) - sqrt(x+h+3)sqrt(x+h+3)sqrt(x+3)h f'(x) = _h 0 sqrt(x+3) - sqrt(x+h+3)hsqrt(x+h+3)sqrt(x+3) Step 3: Multiply the numerator and denominator by the conjugate of the numerator. The conjugate of sqrt(x+3) - sqrt(x+h+3) is sqrt(x+3) + sqrt(x+h+3). f'(x) = _h 0 (sqrt(x+3) - sqrt(x+h+3))(sqrt(x+3) + sqrt(x+h+3))hsqrt(x+h+3)sqrt(x+3)(sqrt(x+3) + sqrt(x+h+3)) Step 4: Simplify the numerator using the difference of squares formula (a-b)(a+b) = a^2 - b^2. The numerator becomes: (sqrt(x+3))^2 - (sqrt(x+h+3))^2 = (x+3) - (x+h+3) = x+3 - x - h - 3 = -h Substitute this back into the expression: f'(x) = _h 0 (-h)/(hsqrt(x+h+3)x+3)(sqrt(x+3) + sqrt(x+h+3)) Step 5: Cancel out h from the numerator and denominator. f'(x) = _h 0 (-1)/(sqrt(x+h+3)x+3)(sqrt(x+3) + sqrt(x+h+3)) Step 6: Substitute h=0 into the expression. f'(x) = (-1)/(sqrt(x+0+3)x+3)(sqrt(x+3) + sqrt(x+0+3)) f'(x) = (-1)/(sqrt(x+3)x+3)(sqrt(x+3) + sqrt(x+3)) f'(x) = (-1)/((x+3)(2sqrt(x+3))) Step 7: Simplify the denominator. Since sqrt(x+3) = (x+3)^1/2, we can write (x+3)sqrt(x+3) as (x+3)^1 (x+3)^1/2 = (x+3)^1 + 1/2 = (x+3)^3/2. f'(x) = (-1)/(2(x+3)^3/2) The derivative of y = (1)/(sqrt(x+3)) using the first principle is: -(1)/(2(x+3)^3/2) Send me the next one 📸