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This requires the chain rule and the product rule to be used to differentiate the function. The substitution u = x² can be used to make this easier. Using this, du/dx = 2x and y = (u-5)e^u<...

To find a local minimum (i.e. a point where the function changes from a negative slope into a positive slope), we first need to find all points where the slope of the function is zero. The first derivativ...

A simple way to prove this is to sub in the values that we are given. so f(x) will represent our equation x^3-3x+1 (that is f(x) = x^3-3x+1)f(-2) = -1 < 0f(-1) = 3 > 0The first thing we notice is th...

Start off with:x²+x=12Subtract 12 from both sides:x²+x-12=0Factorise:(x-3)(x+4)=0Solution is therefore:x=3 or x=-4

Here we have to differentiate a constant raised to the power of a variable. To make it easier, let u=sinx and so our function can now be treated as y=a^u. Remembering that A = e^(LnA), a^u = e^(Ln(a^u)). ...