Question

If f(x) = 4 sin(x) + 2x^x, find f′( 2 ).

Answer 1

I assume you can take the derivative of sin(x)

for x^x write x as
\[ x= e^{\ln(x)} \]
and
\[ x^x= \left(e^{\ln(x)}\right)^x =e^{x\ln(x)} \]

Answer 2

so the derivative is 4 cos(x) + 2(e^(xln(x)))?

Answer 3

@phi

Answer 4

4 cos(x) is ok
\[\frac{d}{dx} e^u = e^u \frac{d}{dx}u \]

I don't see the the derivative of x ln(x)

Answer 5

@phi so it is 4cosx+ e^(xln(x))*(1+ln(x)?

Answer 6

yes. but remember that to find the derivative we used
\[ x^x= e^{x\ln(x)} \]
so you could rewrite the answer as
\[ 4 \cos x + x^x(1+\ln(x)) \]

notice the question says " find f′( 2 ). "
so you need to replace x with 2 and simplify

Answer 7

also, the original equation
f(x) = 4 sin(x) + 2x^x

has a 2 * x^2
you are missing the 2 out front of the 2nd term

Answer 8

evaluate at x=2
\[ 4 \cos x + 2x^x(1+\ln(x)) \]