Question

Find the dy/dx for the following functions:

i) y = x^2Secx
ii) y = 2xe^2x / sqrt x^2 + 3x

Answer 1

i) use product rule
let f = x^2 .............f' = 2x
g = sec(x) ............g' = sec(x)tan(x)
--> f'g + fg'
= 2xsec(x) + x^2sec(x)tan(x)

ii) use quotient rule, then product and chain rules
Let f = 2xe^(2x)
Find f' :
a = 2x .............. a' = 2
b = e^(2x) ..........b' = 2e^(2x)
f' = 2e^(2x) + 4xe^(2x)

Let g = sqrt(x^2 +3x)
Find g':
using chain rule, let u = x^2 +3x, du = 2x +3
d/du sqrt(u) = 1/2sqrt(u)
g' = (2x+3)/2sqrt(x^2+3x)

quotient rule
--> (f'g -fg')/g^2
= [ (2e^(2x) + 4xe^(2x))sqrt(x^2 +3x) - 2xe^(2x)(2x+3)/sqrt(x^2 +3x) ] / (x^2+3x)
=[(2e^(2x) + 4xe^(2x))(x^2+3x) - 2xe^(2x)(2x+3)] / (x^2+3x)sqrt(x^2 +3x)
= [2x^2e^(2x)(2x+5)] / (x^2+3x)sqrt(x^2 +3x)