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First of all, we have to use the product rule, since two things are multiplied together.  The product rule states that d/dx (u*v) = vu' + uv'

Let u = (4x+3)⁵  v = sin(2x)

Now, to ...

x^2+5x+6=0 When solving quadratic equations our goal is to revert back into the form (x+a)(x+b) from the form ax^2+bx+c Thus we need to have a method when completing this: A good method in this case is th...

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.