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# Justifying the power rule

Let's explore the power rule's validity by examining the derivatives of x¹ and x². We'll analyze the slopes of tangent lines for these functions and then see how the power rule provides reasonable results, building our confidence in its usefulness. Created by Sal Khan.

## Want to join the conversation?

• when we say slope =4 what do we actually mean? Because when I think of slope I think of degrees
• When I think of slope, I think of rise/run. You may remember hearing your teacher say "rise over run". For slope = 4, I think to myself slope = 4 = rise/run = 4/1. That tells me for every 4 units up, I go over 1.
• How about if there was a square root of x as the denominator of a fraction? For example: f(x) = 1/โx *Please help, and thank you : ) *
• When taking the derivative of โx, it's a good idea to rewrite it as โx = x^(1/2). So when you have 1/โx, you would write it as x^(-1/2). Now you can just use the power rule:
Derivative of 1/โx = Derivative of x^(-1/2)
=(-1/2)x^(-1/2 -1) = (-1/2)x^(-3/2)
• Given f(x) = x, is f'(0) undefined, as x^0 is defined only if x != 0?
• Ok, I figured out that in this case we need to fall back on the derivative definition:
`f'(x) = lim h->0 (f(x+h)-f(x))/h = lim h->0 (x+h-x)/h = lim h->0 h/h = lim h->0 1 = 1`
This way f'(x) is defined even for x=0
• Does the power rule work with polynomials like 2x^3+4x^2+9x+17?
• There is a theorem that states that the derivative of a sum is equal to the sum of the derivatives. So yes! Just use the power rule on each term separately, and then add them all up.
• I know it late to ask but I just wanted to ask that is f(x) always equal to the y? Even in 3-D?
• In the two dimensional case we typically label the dependent variable Y, meaning it's value is dependent on other variables, in 2D there is only one other variable, usually called X (an "independent variable". There is nothing special about their labels howevere, and you could swap X and Y or even solve for Y for instance x = (y - b)/m for slope intercept and graph that way. The X and Y of the graph are just traditional standards, it is more important to understand how dependent and independent variables relate to eachother.

In three dimensions, if solving for Y, X and Z are would normally both be dependent variables
• So if n is 0 and f(x) = x, would f'(x) = 1/x?
• No. The derivative of x is 1. If f(x) was x^0 then the power rule states that you would pull the 0 out in front and subtract one from the exponent, well, anything multiplied by 0 is still 0.
• This may be a tad off topic... I apologize in advance.

If you take any function f(x) and take the derivative of the derivative of the derivative... etc. Where will you end up? If you keep going to infinity, will the derivative just be a horizontal line?
• Hi Natalie,

Depends on the function. Some of them are interesting for example, Google the derivative of f(x) = e^x

For the types of function that you were likely thinking about I have a few questions that will lead you to the answer.

1) What is the derivative of a constant?

2) Is zero a constant?

Regards,

APD
• After we make the curve on the graph , what does making a seacant line represents ?
I mean like in the above video what is the use of makin a seacant line after makig the curve . The slope of any point can also be calculated without makin the seacant line .
• From what I have learned we would use the Secant line for the average slope of a curve between two points and then use the Tan Line as h->0 to get the instantaneous rate of change of the curve at any given point as h->0 Look at the video for Diveratives and it shows a lot more information about the difference of the secant line and Tan Line.
Also one key note the steps to get the secant line and tan line are very similer the only difference is you take the LIM as h-> 0 to get the tan line.
• How would the power rule apply to a question that asks for you to simplify and answer using a single exponent of something like this
(v^6)^5