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2cos(2x) = 1-2sin(x)

2(1-2sin²(x)) = 1-2sin(x) -- by our trigonometric functions

2-4sin²(x) = 1-2sin(x)

4sin²(x)-2sin(x)-1=0 -- rearrange

sin(x...

Okay so, first of all the product rule in itself is fairly simple. Take the product of a function "u" and a function "v" to be a compound function. You can write this as "uv"...

You would have to rewrite 25 as 5^2 hence 25^x = (5^2)^x = 5^(2*x).

Then 5^(2x) = 5^(4x+1) and therefore 2x = 4x + 1 hence 2x = -1 hence x = -1 / 2. 

Write down the formulas involving cos2x and select the one which involves only cosine, this is because cosine (or derivations of it) is the only trigonometric function in this question. Substitute the cho...

dy/dx = 6x² + 4x + 9

dy = (6x² + 4x + 9) dx 

integrating gives:

y= (6x³/3) + (4x²/2) + 9x + c

y= 2x³ + 2x² + 9...