Find the derivative of the following expression: y=x^3+2x^2+6x+5.

As the expression contains 4 separate terms that are added together, we can treat this as 4 separate differentiations. The general rule for differentiation for a general term bx^n is (bn)x^(n-1). For example, the first differentiation in this question is for x³, which would lead to (31)x^(3-1), as b is just 1, which can be simplified to 3x². Carrying on, we can find the following differentiations, which are:

2x² goes to 4x

6x goes to 6

5 goes to 0

The final differentiation is 0 due to the fact that while it is written in the question as 5, we can actually think of it as 5x⁰. This is due to the fact that anything to the power of 0 is 1, leaving just 5. Using the general rule again, this would give the result of (5*0)x^(0-1) = 0.

The final result is therefore dy/dx = 3x²+4x+6.