Power Rule
What is the derivative power rule?
The derivative power rule allows us to more easily find the derivative of polynomial functions and it is defined as follows:
Power Rule:
If [latex]\Large{f(x) = x^n}[/latex] then [latex]\Large{f'(x) = nx^{n-1}}[/latex]
or
[latex]\Large{\frac{d}{dx}(x^n) = nx^{n-1}}[/latex]
Notice the two different notations used to express the derivative of a function. Essentially, the power rule consists of bringing the exponent down to the front and multiplying it, then subtracting the original exponent by 1.
Let us now use the derivative power rule to find the derivative of the function below:
[latex]f(x) = 3x^8+4x^5-3x^2+2[/latex]
First, the sum and difference rule tells us that we can find the derivative of each term separately. We will use the power rule to find derivative of each term as follows:
[latex]\frac{d}{dx}(3x^8+4x^5-3x^2+2) = \frac{d}{dx}(3x^8) + \frac{d}{dx}(4x^5) – \frac{d}{dx}(3x^2)+\frac{d}{dx}(2)[/latex]
Now, using the power rule, we have:
[latex]\frac{d}{dx}(3x^8) = (8)(3)x^{8-1} = 24x^7[/latex]
[latex]\frac{d}{dx}(4x^5) = (5)(4)x^{5-1} = 20x^4[/latex]
[latex]\frac{d}{dx}(3x^2) = (2)(3)x^{2-1} = 6x^1[/latex]
[latex]\frac{d}{dx}(2) = 0[/latex]
Notice that the derivative of a constant is zero. Now, if we combine all the terms, we have:
[latex]\frac{d}{dx}(3x^8+4x^5-3x^2+2) = 24x^7 +20x^4-6x [/latex]
So, the derivative is [latex]f'(x) = 24x^7+20x^4-6x[/latex]
We have just explored how to use the derivative power rule, as well as the sum and difference rule to find the derivative of a polynomial function. Try it yourself and solve the example below!
Example
Using the power rule, find the derivative of the function:
[latex]\Large{f(x) = \sqrt{x}+4x^7-6x^2}[/latex]
Step 1:
Rewriting, we have: [latex]\large{f(x) = x^{\frac{1}{2}}+4x^7-6x^2}[/latex]
Step 2:
Applying the derivative power rule:
[latex]\large{f'(x) = (\frac{1}{2})x^{\frac{1}{2} – 1}+(7)(4)x^{7-1}-(2)(6)x^{2-1}}[/latex]
Then simplifying, our answer is:
[latex]\large{f'(x) = \frac{1}{2\sqrt{x}} + 28x^6 \:-\: 12x}[/latex]
Step 1:
Rewriting, we have:
[latex]f(x) = x^{\frac{1}{2}}+4x^7-6x^2[/latex]
Step 2:
Applying the derivative power rule:
[latex]\small{f'(x) = (\frac{1}{2})x^{\frac{1}{2} – 1}+(7)(4)x^{7-1}-(2)(6)x^{2-1}}[/latex]
Then simplifying, our answer is:
[latex]f'(x) = \frac{1}{2\sqrt{x}} + 28x^6 \:-\: 12x[/latex]
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