Power Rule for Derivatives

Power Rule for Derivatives

The power rule is one of a number of derivative rules in calculus. The power rule says that
if a function is raised to a power, then the derivative is equal to that function raised to the
same power times the original function's derivative.
For example, suppose we are given a function f(x) and its derivative f'(x), then the power
rule savs:
If f(x) = xn, then
f'(x) = nxn-1
The power rule is a shortcut for computing derivatives of exponents.

So it's possible that you have encountered the fact that if you were to define the derivative
for every single derivative that existed for square functions, you would be in for a very long
year. But what if I said, "hey let's take the derivative of × to the 10th power"?
Of course, there is Pascal's triangle and the rules for multiplying out a binomial. However,
that is a time-consuming process that is not worthwhile.

There are also trig functions and
exponential functions, as well as logarithmic functions so many functions that it would be
more efficient to take their derivative than to use them directly.
So instead of trying to solve the problem in its current form, we'll take a different approach.
We'll take the derivative of the function three and graph it.
The derivative of a horizontal line at any point on the line is 0. The derivative of x is the
slope of a horizontal line tangent to the graph at that point.

Get quality help now

Writer Lyla

Verified

Proficient in: Calculus

5 (876)

“ Have been using her for a while and please believe when I tell you, she never fail. Thanks Writer Lyla you are indeed awesome ”

+84 relevant experts are online

Hire writer

Okay, so now let's take the derivative of x. The slope of a line that is tangent to the graph
at a point is equal to the derivative of the function at that point. If I draw such a line, it has
a slope of 1.
You might have seen this from the example video. We have already done 3x squared,
which gave us 6x, hint. Yes, the derivative of x squared is 2x.
The slope of a line is always changing from positive to negative and back again. The slope
of the line y=2x can be found at the point where its value is 0, which is where x=0.
So what is the pattern? What is the trick? Can you solve for the derivative of x cubed?
You are correct, the derivative of ×4 is 3×2.
A pattern can be difficult to see at first, but if we examine it closely, we can see that the
exponent becomes the coefficient, and then we subtract from the exponent. So this
means we have 4x cubed.
IfI was to formulate a rule, it would be called the power rule for derivatives.
The power rule for derivatives states that if x is raised to the nth power, then the derivative
of x with respect to t equals nxn-1.
For any exponent, even rational expressions, the square root function will work. For
example. If I have 1 over x squared, that can be rewritten as x to the negative 2. The
square root of x is x to the 1/2.
If the function could be written as a power, I could take its derivative by using this rule. If
there were a constant coefficient in front of the X. like 3×2, then I could take the derivative
by multiplying that constant by itself.
In our example, we saw this phenomenon: if you take the derivative of x2, you can factor
out a 3.
The derivative of x2 is 2x times 3, which gives us 6x.
[Graph][Graph][Graph]