The Power Rule — Special Cases
Five special cases of the power rule — a bare variable, a constant, a coefficient out front, a negative exponent, and a fractional exponent — each worked through the same two steps.
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Five special cases of the power rule — a bare variable, a constant, a coefficient out front, a negative exponent, and a fractional exponent — each worked through the same two steps.
A negative exponent, a fraction, a bare constant — the power rule covers all of them. Here are five cases that look like exceptions, each run through the same two steps: bring the exponent down, subtract one.
Write with its exponent shown: . Then apply the rule.
, so the derivative is the constant — the line has slope everywhere.
The coefficient of out front kills the whole term, whatever the exponent does. The derivative of any constant is .
A constant multiplier just rides along. Differentiate the power and keep the attached.
The exponent comes down and multiplies the to give ; the exponent left behind is .
Nothing changes. Bring down, then subtract one from it: .
Watch the signs: subtracting one from a negative exponent makes it more negative. Written as a fraction, .
A root is a fractional exponent: . Bring down, then .
The negative exponent moves into the denominator, and there is written as .