how do I find the derivative of f(x)= 4/ square root of x ? using the limit process?
F(x) = Y = 4/sqrt(X) = 4/X^(1/2),
Y = 4X^(-1/2),
Y'=-2X^(-1/2-1) = -2X^(-3/2= -2/X^(3/2)= -2/sqrt(X^3).
using the limit process how do i find the derivative of f(x)=4/sqrt of x?
f '(x) = lim [4/√(x+h) - 4/√x]/h as h ---> 0
= lim [(4√x - 4√(x+h))/(√x√(x+h))*(1/h)*(4√x + 4√(x+h))/(4√x + 4√(x+h))
= lim [16x - 16(x+h)]/(√x√(x+h)(4√x + 4√(x+h)) * 1/h
= lim (-16h)/(√x√(x+h)(4√x + 4√(x+h)) * 1/h as h ---> 0
= -16/(2x(8√x))
= -1/(x√x) or -1/x^(3/2) or -(x^-(3/2)) or -1/(√x)^3
To find the derivative of the function f(x) = 4/√x using the limit process, we follow these steps:
Step 1: Write the function as a fraction.
f(x) = 4x^(-1/2)
Step 2: Use the definition of the derivative.
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Step 3: Substitute the function into the definition of the derivative.
f'(x) = lim(h→0) [(4(x+h)^(-1/2)) - (4x^(-1/2))] / h
Step 4: Simplify the expression by expanding the numerator.
f'(x) = lim(h→0) [4(x^(-1/2) + h^(-1/2)) - 4x^(-1/2)] / h
Step 5: Combine like terms.
f'(x) = lim(h→0) 4h^(-1/2) / h
Step 6: Simplify further by rationalizing the denominator.
f'(x) = lim(h→0) 4 / (h*(h^1/2))
Step 7: Divide both numerator and denominator by h to simplify.
f'(x) = lim(h→0) 4 / (h^3/2)
Step 8: Apply the limit to find the derivative.
f'(x) = 4lim(h→0) / (h^3/2)
Step 9: Evaluate the limit.
f'(x) = 4 * ∞ / 0^3/2
= 0
Therefore, the derivative of f(x) = 4/√x using the limit process is 0.