limit as delta x approaches 0 when f(x) = 4/ sqr x
Have to use the (F(x+delta x)-f(x))/delta x method
just plug and chug
[f(x+h)-f(x)]/h
4(x+h)^(-1/2) - 4*x^(-1/2)
Now, to evaluate (x+h)^(-1/2), use the binomial theorem:
(a+b)^n = a^n + n*a^(n-1)*b^1 + n(n-1)/2 * a^(n-1)b^2 + ...
(x+h)^(-1/2) = x^(-1/2) + (-1/2)*x^(-3/2)*h + (-1/2)(-3/2)/2 * x^(-5/2)h^2 + ...
(x+h)^(-1/2) - x^(-1/2) = (-1/2)*x^(-3/2)*h + (-1/2)(-3/2)/2 * x^(-5/2)h^2 + ...
divide all that by h to get
(-1/2)*X^(-3/2) + (-1/2)(-3/2)/2 * x^(-5/2)h + ...
= -1/2x^(-3/2) + 3/8x^(-5/2)*h + ...
As h->0, all the terms with h disappear, leaving just
-1/2 x^(-3/2)
multiply that by 4 from the original function, and you get
-2x^-3/2 = -2/√x3
To find the limit as delta x approaches 0 when f(x) = 4/sqrt(x), we can use the limit definition of the derivative.
The formula (f(x + delta x) - f(x))/delta x represents the average rate of change of a function f(x) over a small interval. As delta x gets smaller and approaches 0, this expression represents the instantaneous rate of change of the function at a specific point.
In this case, we want to find the derivative of f(x) = 4/sqrt(x) using the limit definition. So, we need to evaluate the following expression as delta x approaches 0:
(f(x + delta x) - f(x))/delta x
Substitute f(x) into the expression:
= [4/sqrt(x + delta x) - 4/sqrt(x)]/delta x
To simplify this expression, we need to get a common denominator for the two fractions in the numerator. Multiply the first fraction by sqrt(x) / sqrt(x) and the second fraction by sqrt(x + delta x) / sqrt(x + delta x):
= [4(sqrt(x) - sqrt(x + delta x)) / (sqrt(x + delta x) * sqrt(x))] / delta x
Next, simplify the numerator by multiplying through by sqrt(x) and distributing:
= [4sqrt(x) - 4sqrt(x + delta x)] / (delta x * sqrt(x + delta x) * sqrt(x))
Now, divide the numerator by delta x:
= (4sqrt(x) - 4sqrt(x + delta x)) / (delta x^2 * sqrt(x + delta x) * sqrt(x))
Finally, we can take the limit as delta x approaches 0:
lim (delta x -> 0) [(4sqrt(x) - 4sqrt(x + delta x)) / (delta x^2 * sqrt(x + delta x) * sqrt(x))]
Evaluating this limit will give us the derivative of f(x) = 4/sqrt(x) at a given point.