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.

2^-8 = ?

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