Help with derivatives?

Question

So I just started derivatives today in calculus, so we have to find the derivative by using the definition.

p(x)=sqrt(3x)

I'm having trouble, help please!

Answer 1

p(x) = (3x)^1/2 you do know that 3x to the 1/2 power is sqrt of 3x right?

p'(x) = (1/2)(3)(x^(-1/2)) the 1/2 goes in front of 3 and then 1/2 - 2/2 =    -1/2 and then derivitive of 3x is 3 so multiply that too.

p'(x) = (3/2)*3x^(-1/2)

EDIT: if you're gonna use limits, it's gonna take forever and sorry i don't want to do it

Answer 2

f ' (x) = lim (h → 0) (f(x + h) - f(x)) / h

p ' (x) = lim (h → 0) (√(3(x+ h)) - √(3x)) / h

= lim (h → 0) (√(3x+ 3h) - √(3x)) / h

= lim (h → 0) (√(3x+ 3h) - √(3x))(√(3x+ 3h) + √(3x)) / (h√(3x+ 3h) + √(3x)))

= lim (h → 0) (3x+ 3h - 3x) / (h√(3x+ 3h) + √(3x)))

= lim (h → 0) 3h / (h√(3x+ 3h) + √(3x)))

= lim (h → 0) 3 / (√(3x+ 3h) + √(3x))

= 3 / (√(3x) + √(3x))

= 3/(2√(3x))

Answer 3

The first thing we can do to make the problem easier is to write the square root as a polynomial to the power of 1/2.  

p(x) = (3x)^(1/2)

When we take the derivative, the first thing we do is multiply the whole equation by the exponent.  We then subtract 1 from the exponent itself.  

= (1/2)(3x)^(-1/2)

The last step is to multiply the whole thing by the derivative of (3x).  This is necessary because of the chain rule-- if you are differentiating anything but a simple "x," you need to multiply the equation by the derivative of "what's inside [the parenthesis]."  The derivative of 3x is 3, so we multiply by 3.

= (3/2)(3x)^(-1/2)

And there you go.  Look up "chain rule" on wikipedia if you still have troubles.