Been working on this for too long and the answer I get is Always wrong! :/ Q: f(x)=3sin(x)+4x^x, Find f'(3)

Answer 1

This is tough because the derivative of X^X is not found by using the general power rule (i.e. the way the derivative of X^2 is 2X).

You need to find it implicitly using logarithms.

The derivative of 4X^X = 4X^X(ln(x) + 1)

The question asks you to take the 1st derivative and evaluate it at 3.

so f'(x) = 3 Cos(X) + 4X^X(ln(x) +1)

at 3...

3 Cos(3) + 108(ln(3) + 1)

See if that works.

Answer 2

BJ 's answer is correct.

A common error is to not pay attention to the hypotheses of the various differentiation rules in a calculus course. In particular , the "power rule"

(x^n)'=nx^(n-1)

is true when n is a constant. the expression x^x is of the form "a function raised to a function", so the power rule does not apply. One needs either to rewrite the expression in terms of the exponential function or use implicit differentiation of y=x^x after taking natural logs of both sides and simplifying the right side.

Answer 3

I was trying to figure out the derivative, but I'm not sure how to do the 4x^x part. I'm in calculus right now too, I guess we haven't gotten to that part yet...but, I plugged in the derivative formula in my TI-84, and the answer should be 223.68...sorry I couldn't help with the equation though.

Oh, by the way, if you use a TI-84, too, here's how to graph the derivative of a function without knowing the f' equation.

Plug in your original function in y1. Then in y2, do the following:

press MATH 8. then VARS, go right to Y-VARS then press 1 ENTER. then type (X),X,X)

exactly how I typed it, commas and all.

To check to see if the function you come up with for the derivative is right, plug it into y3, and then go to the table to see if the values of y2 and y3 are the same. If they are, your function is right, and if not, try again!

Answer 4

All you have to do is plug 3 in for every x.

Answer:f(3)=3sin3+4(3)^3
=3sin3+1728
=

Answer 5

Final answer: 3*cos(x)+4*x^x*(ln(x)+1)
Some notes: x^x = e^x*ln(x) and derivative of x^x = x^x*(ln(x)+1).

Answer 6

I'm not understanding all the signs you used; but have you gotten 45 as an answer yet?

Answer 7

f'(x) = 3cos(x) + 4x(4x)^(x-1) (chain rule)

f'(3) = 3cos(3) + 1728

Answer 8

d[f(x)]/dx = d[3sin(x)+4x^x]/dx

d[f(x)]/dx = d[3sin(x)]/dx+d[4x^x]/dx

d[f(x)]/dx = 3d[sin(x)]/dx+4d[x^x]/dx

d[f(x)]/dx = 3[cos(x)]+4[x^x(ln(x)+1)]

d[f(3)]/dx = 3cos(3)+4•3³ln(3+1)

d[f(3)]/dx = 3cos(3)+108ln4 (exact answer)

Answer 9

ok i got
3cos3+108
evaluate cos3, i dont have a calculator right now...
i think that should be right