A curve has equation y = 4x + 1/(x^2) find dy/dx.

As in every case dy/dx can be found by differentiating each term individually with respect to x.

Let's first tackle the 4x term.

As always the derivative can be found by multiplying the term by the power of x and reducing the power of x by 1. i.e. ax^b -> abx^b-1

In this case b is simply equal to 1 because x=x¹. Therefore the derivative of 4x with respect to x is given by 14x⁰ = 41 = 4.

Next let's tackle the 1/x² term.

In order to use the same method as previously we must first write 1/x² as a power of x i.e. in the familiar form ax^b.

From the laws of indices: recall that x^-b=1/x^b. In this example b is simply equal to 2 so 1/x²=x^-2.

Now that we have written the term in the form ax^b we can apply the same method as previously i.e.   x^-2-> -2x^-3.

Finally collecting both the terms we have arrived at the result that dy/dx is -2x^-3+4.