# How do you find f'(x) using the definition of a derivative for #f(x)= 6 x + 2sqrt{x}#?

The derivative is

#f'(x)=lim_(h->0)(f(x+h)-f(x))/h=lim_(h->0)(6(x+h)+2sqrt(x+h)-6x-2sqrtx)/h=lim_(h->0) 6+2*(sqrt(x+h)-sqrtx)/h= 6+2lim_(h->0)(sqrt(x+h)-sqrtx)/h= 6+2lim_(h->0)h/(h(sqrt(x+h)+sqrtx))=6+2*1/(2sqrtx)=6+1/sqrtx#

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To find ( f'(x) ) using the definition of a derivative for ( f(x) = 6x + 2\sqrt{x} ), we use the definition of the derivative:

[ f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} ]

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

[ f(x + h) = 6(x + h) + 2\sqrt{x + h} ]

Now, subtract ( f(x) ):

[ f(x + h) - f(x) = 6(x + h) + 2\sqrt{x + h} - (6x + 2\sqrt{x}) ]

[ = 6x + 6h + 2\sqrt{x + h} - 6x - 2\sqrt{x} ]

[ = 6h + 2\sqrt{x + h} - 2\sqrt{x} ]

Now, divide by ( h ):

[ \frac{{f(x + h) - f(x)}}{h} = \frac{{6h + 2\sqrt{x + h} - 2\sqrt{x}}}{h} ]

[ = \frac{{6h}}{h} + \frac{{2\sqrt{x + h} - 2\sqrt{x}}{h}} ]

[ = 6 + \frac{{2\sqrt{x + h} - 2\sqrt{x}}{h}} ]

Now, take the limit as ( h ) approaches ( 0 ):

[ \lim_{{h \to 0}} \left( 6 + \frac{{2\sqrt{x + h} - 2\sqrt{x}}{h}} \right) ]

[ = 6 + \lim_{{h \to 0}} \frac{{2\sqrt{x + h} - 2\sqrt{x}}{h}} ]

To evaluate the limit, we can use the definition of the derivative of ( \sqrt{x} ):

[ f'(x) = 6 + \lim_{{h \to 0}} \frac{{2}}{h} (\sqrt{x + h} - \sqrt{x}) ]

[ = 6 + 2 \lim_{{h \to 0}} \frac{{\sqrt{x + h} - \sqrt{x}}{h}} ]

[ = 6 + 2 \left( \frac{{d}}{{dx}} \sqrt{x} \right) ]

[ = 6 + 2 \left( \frac{{1}}{{2\sqrt{x}}} \right) ]

[ = 6 + \frac{{1}}{{\sqrt{x}}} ]

Therefore, ( f'(x) = 6 + \frac{{1}}{{\sqrt{x}}} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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