Question

one more . What is the derivative of f(x) = ln(x^e +e^x)

Answer 1

The derivative of ln (u) = (i/u)(du) or du/u

So the derivative will be (1/(x^e + e^x)) (ex^(e-1) + e^x)

Answer 2

\(\sf \color{}{\frac{1}{x^e+e^x}(e^x)}\)

Answer 3

Oops, forgot the \(\sf \color{}{ex^{e-1}}\)

Answer 4

can you walk thru the steps a bit

Answer 5

U is (x^e + e^x)

Answer 6

The derivative will be 1/u or 1/(x^e + e^x)
multiplied by the derivative of u.

The derivative of u is the derivative of (x^e + e^x).

The derivative of x^e is tricky...it is like x^constant..since e is a constant. so the derivative of x^e is ex^(e-1); and the derivative of e^x is e^x.


Think of x^e like x^4. The derivative of x^4 is 4x^(4-1) or 4x^3. Similarly, the derivative of x^e is e(x^(e-1)).

Answer 7

Makes sense?

Answer 8

you are awesome... got it.. thanks

Answer 9

Very welcome.

Answer 10

The tricky part is the part when you need the derivative of x^e. Many students will get that part wrong.

Answer 11

It should be crystal clear that the derivative is e(x^(e-1)).

Answer 12

so if you solve for f'(1) what would you get?

Answer 13

Lets see...you would subsitute x = 1 into the derivative, and you'll get.......

Answer 14

(1 + (1/e)) (2e) or 2e + 2 or 2(e+1). Agree?

Answer 15

I got 2e/e+1

Answer 16

Let me double-check....

Answer 17

My answer is correct...let's do it together.

1^e = 1
e^1 = e
1^(e-1) = 1

Answer 18

Agree?

Answer 19

1^(e-1)=1?

Answer 20

1^any number = 1

e-1 is just a number

Answer 21

ok

Answer 22

true..

Answer 23

we are goo

Answer 24

good

Answer 25

so do the arithmetic again carefully..you will see my answer is correct.

If you still dont have what I got, then we will have to work it step-by-step

Answer 26

its fine... thenks

Answer 27

poor spelling

Answer 28

Is it fine? Are you 100% convinced that f ' (1) = 2(e+1)???

Answer 29

we have an e^e-1 what is this simplified to

Answer 30

thats 0...because x^a multiplied by x^b = x^(a+b)
so e^1 times e^(-1) = e^0 = 1

Answer 31

add the exponents.

Answer 32

e is the base;

Answer 33

one exponent is 1; the other exponent is -1;

1 + (-1) = 0

e^0 = 1

any number to the 0 power = 1.

Answer 34

but its not that... it is e-1 so actually 1.7

Answer 35

e^e-1

Answer 36

e(x^(e-1)) means you do x^(e-1) multiplied by e

Answer 37

just like 4x^3 means x^3 multiplied by 4

Answer 38

You are putting in x = 1, so e(x^(e-1)) = e(1^(e-1)) = e(1) = e

Answer 39

so than it would be 2e on the numerator

Answer 40

agree...and the denominator, you get what?

Answer 41

e+1

Answer 42

agree...so you get (2e/(e+1)) ...you are correct..I must have made an arithmetic error along the way.

Answer 43

it happens.. thanks

Answer 44

With so many x's and e's..some error may occur if not extra careful.

Answer 45

I am aware

Answer 46

Good luck.

Answer 47

But, you know something...on an exam..such an error, MAY result in a 1-point loss...as thats a careless arithmetic..whats important is the calculus.

Answer 48

multiple choice test answers... worse

Answer 49

Of course, we want the exact value of f ' (1).

Answer 50

True...but I cant believe such a multiple-choice question asking for f ' (1). No professional exam would include this question asking for f ' (1). The derivative, yes...not f ' (1).

Answer 51

We want to know if the student knows the calculus; not if they will make a careless arithmetic error.