If f(x) = 2 √x, use the definition of derivative to find f′( x )

To find the derivative of the function f(x) = 2 √x using the definition of derivative, we need to utilize the limit definition of the derivative. The derivative of a function f(x) at a point x is defined as:

f'(x) = lim (h->0) [f(x + h) - f(x)] / h

Applying this definition to our function f(x) = 2 √x, we have:

f'(x) = lim (h->0) [f(x + h) - f(x)] / h

First, let's find f(x + h):

f(x + h) = 2 √(x + h)

Next, we substitute f(x + h) and f(x) back into the derivative equation:

f'(x) = lim (h->0) [2 √(x + h) - 2 √x] / h

Now, let's simplify the expression by multiplying both the numerator and denominator by the conjugate of the numerator, which is [2 √(x + h) + 2 √x]:

f'(x) = lim (h->0) [(2 √(x + h) - 2 √x) * (2 √(x + h) + 2 √x)] / [h * (2 √(x + h) + 2 √x)]

Expanding the numerator:

f'(x) = lim (h->0) [4(x + h) - 4x] / [h * (2 √(x + h) + 2 √x)]

Simplifying the numerator:

f'(x) = lim (h->0) [4x + 4h - 4x] / [h * (2 √(x + h) + 2 √x)]

f'(x) = lim (h->0) [4h] / [h * (2 √(x + h) + 2 √x)]

Now, we can cancel out the common factors of h:

f'(x) = lim (h->0) 4 / (2 √(x + h) + 2 √x)

Finally, we take the limit as h approaches 0:

f'(x) = 4 / (2 √x + 2 √x)

f'(x) = 4 / (4 √x)

Simplifying the expression further:

f'(x) = 1 / √x

Therefore, the derivative of f(x) = 2 √x is f'(x) = 1 / √x.

f(x)=2√x

f'(x)
=lim h->0 2(√(x+h)-√x)/h
=lim h->0 2((x+h)-x)/(h*(√(x+h)+√x))
=lim h->0 2h/(h*(√(x+h)+√x))
=lim h->0 2/(√(x+h)+√x)
=2/(2√x)
=1/√x