Question

how do you find the tangent line of sqrt(x)*f(x)?

Answer 1

\[\Large \color{royalblue}{\text{Welcome to OpenStudy! :)}}\]

Answer 2

So are we looking for the slope of the line tangent to the function at a specific x value?
Or just the generalized equation which gives us these tangent lines?

Are you currently working with the `Limit Definition of the Derivative` ?
\[\Large \lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]

Answer 3

the full question says the tangent line of a graph of a function f at the point x=3 is given by the equation y=3*x+1. find the tangent line to the graph of the function sqrt(x)*f(x) at the point x=3. put your answer in y=m*x+b

Answer 4

Have you learned `Power Rule` and `Product Rule` and all those fun things yet? :)

Answer 5

yes (:

Answer 6

Ok so they're trying to get you used to what a derivative represents.
They don't tell us what f(x) actually is.

But they give us a tangent line for the function at x=3.
The `slope` of the tangent line represents the `value` of the derivative at that point.
\[\Large y=\color{royalblue}{3}x+1 \qquad\to\qquad f'(3)=\color{royalblue}{3}\]

Answer 7

Hmm what else can we get from this... let's seeeee

Answer 8

i do remember that, so that is good!

Answer 9

So we have some other function, let's call it g(x).\[\Large g(x)\quad=\quad \sqrt{x}\cdot f(x)\]Taking the derivative, applying the product rule gives us,\[\Large g'(x)\quad=\quad \color{#CC0033}{\left(\sqrt x\right)'}f(x)+\sqrt x \color{#CC0033}{f'(x)}\]

Answer 10

Remember how to differentiate square root? :o

Answer 11

x^(1/2)?
is that what you're saying?

Answer 12

That's a way that we can rewrite the sqrt, it allows us to take the derivative easier.
Can you find the derivative of it from there?

Answer 13

well we still don't know what f(x) is?

Answer 14

Hmm that's true, we don't.
That might cause a problem for us :u ...

Let's deal with this part first though,\[\Large \color{#CC0033}{\left(\sqrt x\right)'}\quad=\quad ?\]

Answer 15

(1/2)*x^(-1/2)

Answer 16

Yes ok good :)\[\Large \color{#CC0033}{\left(\sqrt x\right)'}\quad=\quad \frac{1}{2\sqrt x}\]

Answer 17

Hmm we still need f(x) don't we? Hmm lemme think a sec.

Answer 18

So our tangent line to f(x) at x=3 is given by y=3x+1.

The tangent line `touches` the curve f(x) at x=3, right?
Example:^That's just an example, but see how the tangent line is `touching` the curve at one point?