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Let f ( x ) = 3/ x + ln ⁡ ( b x ), where b is a positiveconstant. Find f ′ ( x )....

Question

Let f ( x ) = 3/ x + ln ⁡ ( b x ), where b is a positiveconstant. Find f ′ ( x ).

Let f ( x ) = 3/ x + ln ⁡ ( b x ), where b is a positive constant. Find f ′ ( x ).



Answers

Determine $f^{\prime}(x)$. $$f(x)=\ln 3 x$$

So this problem wants to take the derivative of the function eight times extra cubed plus B X squared plus c expose D or A, B and C and D are all constants. So this is a polynomial with a bunch of power functions. So we'll be using the power rule. Take the derivative its function and the power rule says that we want take a derivative of power function will have to bring down the exponent and multiplied by X city and minus one. So we'll be using this a lot. And the constants A, B, C, NT are all coefficient constants. Will A, B and C your crawfish and constant soul. Do you believe those things alone as we solve the problem? So a since I was just a constant in front of X cubed we'll leave that alone so we just don't differentiate, execute and using the power rule that becomes three x square. Next is B X squared. You know that B is a coefficient constant, so we'll leave that alone. And the drill of X squared by the power role is two times X since two minus one is just one. So just accept one power, but we can just think about it. Just acts and see time on C Times X on the derivative of X is just one. So we just getting see as a constant and finally the derivative of D, which is a constant without an X variable, so that just become zero. Simplifying this, we'll get a form of X equals two a. Three a x squared was to be times X plus c, and this is your final answer.

In this video we will find the value of F B minus F. Of a. Given that f of X is equal to negative X cubed over three plus three. X plus C. A is equal to one and B is equal to two. So the first thing we can find is FFB by plugging in 24 X. Which gets us -2 cubed plus or over three plus three times two plus C. This is equal to negative 8/3 plus six plus C. Which is negative 8/3 plus 18/3 plus C. Or 13th plus C. Next we can find F. F. A. By plugging one in for X which gets us a negative one cube over three plus three times one plus C. Which is negative one third plus three plus C. Or negative one third plus nine thirds plus C, Which is 8/3 plus e. And then we can put F. B. And F. Of A. Into the equation of b minus half of a environment that is 13th will c minus eight thirds pussy, which is 13th minus eight thirds plus c minus C. And the seas cancel out, and we get two thirds, and so that is our final answer.

We need to find the derivative off given function F or fax equals three into natural log off X square plus one fo fax can be represented by why so if y equals f or fax, the given function can be represented by the expression why equals three natural log off X square plus one We substitute X square plus one by U So u equals x square plus one. Why can then be written us? Why equals three into natural? Log off you next. Finding the derivatives of each of these expressions we get Do you by D X equals two x d y by D. U yeah, equals three Divided by you The derivative off natural log A few is always one by you. F Dash affects can be written as D y by D. X, which can be written as d y by d you into do you by D X by chain rule off derivatives Putting in the values we get three by you into two x on simplification and putting in the value off you. We get six x divided by X square plus one

Here we are looking for F X equal to three times line X minus for also equal to three. My class four on both sides. That's I know for no negative for any more. So three plus four is seven line X d Read about three. Okay, three line X divided by three. We have length like eggs. Sorry. Have seven. Divided by three s where all we're used to property the e to the power line eggs You go to X So we have e to the power line. Thanks. And he to the power seven divided by three. So we have X on the left to the power seven. Divide about three. Answer. Thanks for watching.


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