Differentiate 8x^4 + 2x^2 + 10
Differentiating is a process to turn one expression into another, which has many uses in maths. Our expression is made up of three parts: 8x4, 2x2 and 10. We differentiate each part seperately. The process of differentiating for polynomial expressions (polynomial just means expressions made up of x's that have powers e.g. x4) goes like this:
-
take the power of the x and make it the new x term's coeffecient (put it in front of the x, so effectively you are multiplying the new term by it).
-
reduce the power of x by 1, to give the x a new power
-
simplify by multiplying any coefficients
This may sound confusing, but it is simple in practice. So for our expression, take the first term 8x4. The first step is to multiply the new term by the power, so we write 48x, we leave the new power of x for now. Note that the 8 stays the same, because coefficients of x are always unchanged. Then we reduce the power of x by 1, so 4 minus 1 = 3. So we now have 48x3. Then we simplify, by multplying the two coefficients, 4*8 = 32. So the first term is 32x3.
We repeat this process for the next term, so 2x2 becomes 4x. Then the last part, +10, becomes 0 so we do not write anything. The rule is, if a term is just a number on its own (e.g. +10) it becomes 0. (This is because we can imagine the +10 having an x0 attached to it, which equals 1. Then when we multiply by the power of x, we times by 0 so we get 0 as the answer.) Therefore overall, we are left with: 32x3 + 4x. This is the correct answer.
