Find the derivative by the limit process.
6/√(8x)
To find the derivative of the given function, we can use the limit process, also known as the definition of derivative. Here's how to do it step by step:
Step 1: Start by writing down the definition of the derivative:
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
Step 2: Substitute the given function into the definition:
f(x) = 6 / √(8x)
f'(x) = lim(h→0) [6 / √(8(x + h)) - 6 / √(8x)] / h
Step 3: Simplify the expression:
To simplify the expression, we need to rationalize the denominator. We can do this by multiplying the numerator and denominator by the conjugate of the denominator.
f'(x) = lim(h→0) [6 / √(8(x + h)) - 6 / √(8x)] * [√(8(x + h)) + √(8x)] / [√(8(x + h)) + √(8x)] / h
Step 4: Expand the numerator:
f'(x) = lim(h→0) [6 * (√(8(x + h)) + √(8x)) - 6 * (√(8(x + h)) - √(8x))] / [h * (√(8(x + h)) + √(8x))]
Step 5: Simplify the expression further:
f'(x) = lim(h→0) [6√(8(x + h)) + 6√(8x) - 6√(8(x + h)) + 6√(8x)] / [h * (√(8(x + h)) + √(8x))]
Step 6: Cancel out the common terms:
f'(x) = lim(h→0) [12√(8x)] / [h * (√(8(x + h)) + √(8x))]
Step 7: Further simplify the expression:
f'(x) = lim(h→0) [12√(8x)] / [h√(8(x + h)) + h√(8x)]
Step 8: Factor out a common factor from the denominator:
f'(x) = lim(h→0) [12√(8x)] / [h(√(8(x + h)) + √(8x))]
Step 9: Cancel out the common factor in the numerator and denominator:
f'(x) = lim(h→0) [12] / [√(8(x + h)) + √(8x)]
Step 10: Evaluate the limit as h approaches 0:
f'(x) = 12 / [√(8x) + √(8x)]
= 12 / (2√(8x))
= 6 / √(8x)
Therefore, the derivative of the function 6 / √(8x) with respect to x is 6 / √(8x).