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If we let y = lnx, we then know that x = e^y. By differentiating both sides of this equation with respect to y we get:dx/dy = e^y, as the exponential function differentiates to itself ...

We start with sec^2 x + 2tanx = 0 for 0 ≤ x ≤ 2π
Note that sec^2 x = tan^2 x +1
tan ^2 x + 2tan x +1 = 0
(tan x +1)^2=0
tan x = -1
x=3π/4 or x=7π/4

In a proportional equation, you represent the factor of proportionality as k. In this case it means that y=kx^2. In the question you're given a number for both y and x. Sub these in in order to solve for ...

No atoms are lost or made during a chemical reaction, so the total mass of the products is equal to the total mass of the reactants.

Take out all common factors, in this case only x. This leaves you with x(1-4x^2).Within the brackets you now have a quadratic equation which you can factorise. You need to find factors of -4x^2 that plus ...