Find the first derivative given that,

Question

Find the first derivative given that,

f (x) = 5x

√2x + 3

Answer 1

f(x) = 5x/√(2x+3) = g(x)/h(x)

f'(x) = (g'(x)h(x)-g(x)h'(x))/h²(x)

g'(x) = 5

h'(x) = 1/√(2x+3)

h²(x) = 2x+3

f'(x) =(5√(2x+3)-5x/√(2x+3))/(2x+3) = (5(2x+3)-5x)/(2x+3)^3/2 = (5x+15)/(2x+3)^3/2

Answer 2

I will assume you mean:

You must use the quotient rule

(u/v)' = (u'v - uv')/v²

Where, in this case:

u = 5x meaning u' = 5

v = √(2x + 3) = (2x + 3)^1/2

Using the power rule and chain rule, v' = 1/2(2x + 3)^-1/2(2) = (2x + 3)^-1/2 = 1/√(2x + 3)

And v² = (√(2x + 3))² = 2x + 3

So, put these all together using (u/v)' = (u'v - uv')/v²

To simplify, you need to get a common denominator in the numerator and that will result in a division of fractions which you will simplify using the "keep, change, flip" rule.