In calculus what is the power rule?

Answer 1

I've heard what I'm going to tell you called the "power rule" before. It's the only thing I've ever heard referred to as the power rule.

It's simply the way to find the derivative of a polynomial function, something of the form:

y=axn

where a and n are any real number

The power rule states that to find the derivative (y-prime (y')) of any function of this type, you simply follow this format:

y'=an*xn-1

following this format to higher derivatives:

y''=(a)(n)(n-1)xn-2

y'''=(a)(n)(n-1)(n-2)xn-3

In written terms, you simply bring down the current exponent in front of the variable and lower the exponent by one to make the derivative.

For three examples:

y=6x3

y'=(6)(3)x3-1=18x2

y''=(18)(2)x2-1=36x1=36x

y'''=(36)(1)x1-1=36x0=36

y=7x-3=7/x3

y'=(7)(-3)x-3-1=-21x-4

and so on...

y=4sqrt(x)=4x1/2

y'=(4)(1/2)x(1/2)-1=2x-1/2

and so on...

The last two examples were to show that this method still works for negative exponents, fractional exponents...any exponent that is a real number.

This does not apply only to functions who are purely polynomial. Any component of a larger function that is polynomial can be derived in this way (as long as the component is added or subtracted; division or subtraction brings the product rule into play).

For example:

y=2x2+cos(x-2)

The component "2x2" has a derivative of 4x due to the power rule.

For the record:

y'=4x-sin(x-2)