How do you use the definition of the derivative to find the derivative of the function $f (x) = 6 x + 2 \sqrt{x}$ $f(x) = 6 x + 2sqrt{x}$ ?

Question

2328 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-09T11:12:16

Please look below.

Defintion of a derivative:

$f'(x) = lim_(hrarr0)(f(x+h)-f(x))/h$

$f(x) = 6x + 2sqrtx$
$f(x+h) = 6(x+h) + 2sqrt(x+h)$

$f'(x) = lim_(hrarr0)(6(x+h) + 2sqrt(x+h)-(6x + 2sqrtx))/h$

$f'(x) = lim_(hrarr0)(6h + 2(sqrt(x+h) - sqrt(x)))/h$

$f'(x) = 6+ 2lim_(hrarr0)(sqrt(x+h) - sqrt(x))/h$

$f'(x) = 6+ 2lim_(hrarr0)(sqrt(x+h) - sqrt(x))/h xx (sqrt(x+h) + sqrt(x))/(sqrt(x+h) + sqrt(x))$

$f'(x) = 6+2lim_(hrarr0)(x+h - x)/(h(sqrt(x+h) + sqrt(x)))$

$f'(x) = 6+2lim_(hrarr0)(h )/(h(sqrt(x+h) + sqrt(x)))$

$f'(x) = 6+2lim_(hrarr0)(1 )/(sqrt(x+h) + sqrt(x))$

$f'(x) = 6 + 2 xx 1/(sqrtx+sqrtx)$

$f'(x) = 6 + 1/sqrtx$

How do you use the definition of the derivative to find the derivative of the function f(x) = 6 x + 2sqrt{x}f(x)=6x+2√xf(x) = 6 x + 2sqrt{x}?