(1 pt) If f(x) = 2 √x, use the definition of derivative to find f′( x ).
Anyone?
See
http://www.jiskha.com/display.cgi?id=1287702357
To find the derivative of f(x) = 2√x using the definition of the derivative, we need to recall the definition:
f′(x) = lim(h→0) [f(x+h) - f(x)] / h
Let's substitute f(x) = 2√x into the definition:
f′(x) = lim(h→0) [f(x+h) - f(x)] / h
= lim(h→0) [2√(x+h) - 2√x] / h
Now, let's simplify by rationalizing the numerator:
f′(x) = lim(h→0) [2√(x+h) - 2√x] / h * [(2√(x+h) + 2√x) / (2√(x+h) + 2√x)]
= lim(h→0) [2(x+h) - 2x] / [h(2√(x+h) + 2√x)]
Simplifying further, we get:
f′(x) = lim(h→0) [2x + 2h - 2x] / [h(2√(x+h) + 2√x)]
= lim(h→0) [2h] / [h(2√(x+h) + 2√x)]
= lim(h→0) 2 / [2√(x+h) + 2√x]
The h in the numerator and denominator cancel out, and we are left with:
f′(x) = 2 / [2√(x+h) + 2√x]
Finally, as h approaches 0, we can substitute h = 0 into the equation:
f′(x) = 2 / [2√x + 2√x]
= 2 / (4√x)
= 1 / (2√x)
Therefore, the derivative of f(x) = 2√x is f′(x) = 1 / (2√x).