How solve it? Topic: DERIVATE

Please I need help I do not know how to solveenter image source here

2 Answers
Jun 1, 2018

f(x)=sqrt(2x)

First, let's rewrite the square root as a 1//2 power.

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

Now, we need to recognize that these can be split up as a constant and a variable function.

f(x)=2^(1/2)*x^(1/2)

When we differentiate, multiplicative constants like the 2^(1/2) here simply stay on the "outside," that is, we don't do anything to them.

To differentiate x^(1/2), we use the power rule, which says that d/dxx^n=nx^(n-1).

Then, we see that:

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

Now simplifying:

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

f'(x)=1/(2^(1/2)*x^(1/2))

f'(x)=1/sqrt(2x)

So at x=5/3, the derivative is equal to:

f'(5/3)=1/sqrt(2*5/3)=1/sqrt(10/3)=sqrt(3/10)

Jun 1, 2018

f'(x)=2/[2sqrt(2x)]=1/sqrt(2x)
f'(5/3)=1/sqrt(10/3)=sqrt3/sqrt10

Explanation:

show below:

f(x)=sqrt(2x)

f'(x)=2/[2sqrt(2x)]=1/sqrt(2x)

The derivative at x=5/3 equal

f'(5/3)=1/sqrt(10/3)=sqrt3/sqrt10

"Note that"

color(red)[y=sqrtx]

color(red)[y'=1/[2sqrtx]*x']

color(red)[sqrt[a/b]=sqrta/sqrtb]