Derivative problem?

Question

I have a problem for class in which I have to derive the following:

f(x)=x sqrt(5-x)

I know I have to use Chain Rule, but I can't make the answer match the one in the back of the book. I am so far off, it's not even funny.

The answer given is (10-3x)/2( sqrt(5-x)).

How in the world do I get that?

Answer 1

d(x √(5-x))/dx

Use the product rule:

d(x)/dx √(5-x) + x d(√(5-x))/dx

Use the power rule and chain rule:

√(5-x) + x 1/(2√(5-x)) d(5-x)/dx

Simplify:

√(5-x) - x/(2√(5-x))

Rewrite √(5-x) to have a common denominator:

2(5-x)/(2√(5-x)) - x/(2√(5-x))

Combine like terms:

(2(5-x)-x)/(2√(5-x))

Simplify:

(10-3x)/(2√(5-x))

And we are done.

Answer 2

f(x) = x sqrt(5 - x)

Expressing sqrt(5 - x) as (5 - x)^(1/2),

f(x) = x (5 - x)^(1/2)

Product rule.

f'(x) = (5 - x)^(1/2) + x(1/2)(5 - x)^(-1/2)(-1)

Use the (-1) to make the plus into a minus, and, since
(5 - x)^(-1/2) has a negative exponent, change it to a positive exponent by putting 1 over it.

f'(x) = (5 - x)^(1/2) - x(1/2)(1/(5 - x)^(1/2))

f'(x) = (5 - x)^(1/2) - x/[2(5 - x)^(1/2)])

Now, make a common denominator of 2(5 - x)^(1/2) per usual methods.

f'(x) = [ (5 - x)^(1/2) (5 - x)^(1/2) (2) - x ] / [2(5 - x)^(1/2)]

Simplifying,

f'(x) = [ (5 - x)(2) - x ] / [2(5 - x)^(1/2)]

f'(x) = [10 - 2x - x] / [2(5 - x)^(1/2)]

f'(x) = [10 - 3x] / [2(5 - x)^(1/2)]

Reverting (5 - x)^(1/2) back to sqrt(5 - x),

f'(x) = [10 - 3x] / [2sqrt(5 - x)]

Answer 3

whats a sqrt?