What is the 𝑑𝑦/𝑑𝑥 and simplify the result

Question

Solve for 𝑑𝑦/𝑑𝑥 and simplify the result.

a. 𝑦 = (2𝑥 + 3)³ (3𝑥 − 2)⁴

b. 𝑦 = ( 𝑥² + 1/2𝑥 − 1 )³

c. 𝑦 = 𝑒 𝑥√4 − 𝑒^x

d. 𝑦 = 1/sin𝑥 cos𝑥

Answer 1

y = (2x+3)^3(3x-2)^4

dy/dx = y' = (2x+3)^3(4)(3x-2)^3(3) + (3x-2)^4(3)(2x+3)^2(2)

=12(2x+3)^3(3x-2)^3 + 6(3x-2)^4(2x+3)^2

derivative of a product = 1st term times derivative of the 2nd, plus 2nd times derivative of the 1st

y=(x^2+x/2 -1)^3 use the chain rule

dy/dx = y'= 3(x^2+ x/2 -1)^2(2x+ 1/2)