Find the derivative by the limit process.
6/√(8x)
To find the derivative of the function 6/√(8x) using the limit process, we need to apply the definition of the derivative.
The definition of derivative states that the derivative of a function f(x) at a point c is given by the following limit:
f'(c) = lim(h→0) [f(c + h) - f(c)] / h
In this case, our function is f(x) = 6/√(8x), and we want to find the derivative at a particular point c.
Step 1: Start by writing the definition of the derivative:
f'(c) = lim(h→0) [f(c + h) - f(c)] / h
Step 2: Substitute our function f(x) into the formula:
f'(c) = lim(h→0) [(6/√(8(c + h))) - (6/√(8c))] / h
Step 3: Simplify the expression inside the limit:
f'(c) = lim(h→0) [6/√(8(c + h)) - 6/√(8c)] / h
Step 4: Rationalize the denominators:
f'(c) = lim(h→0) [6(√(8c) - √(8(c + h))) / √(8c)√(8(c + h))) / h
Step 5: Combine the terms in the numerator:
f'(c) = lim(h→0) [6(√(8c) - √(8(c + h))) / (√(8c)√(8(c + h))) / h
Step 6: Simplify the expression in the numerator:
f'(c) = lim(h→0) [6(√(8c) - √(8(c + h))) / (√(8c)√(8(c + h))) / h]
Step 7: Combine the fractions:
f'(c) = lim(h→0) [6(√(8c) - √(8(c + h))) * h / (√(8c)√(8(c + h)))]
Step 8: Distribute the h:
f'(c) = lim(h→0) [6h(√(8c) - √(8(c + h))) / (√(8c)√(8(c + h)))]
Step 9: Cancel out the h in the numerator and denominator:
f'(c) = lim(h→0) 6(√(8c) - √(8(c + h))) / (√(8c)√(8(c + h)))
Step 10: Take the limit as h approaches 0:
f'(c) = 6(√(8c) - √(8(c + 0))) / (√(8c)√(8(c + 0)))
f'(c) = 6(√(8c) - √(8c)) / (√(8c)√(8c))
f'(c) = 6(0) / (2√(2c) * 2√(2c))
f'(c) = 0
Therefore, the derivative of the function 6/√(8x) is 0.