“I’ve got the power!” –Snap (German band)
If you apply the definition of the derivative to several functions, you’ll see:
[latex]\begin{array}{cc} \frac{d}{dx} x^0 & 0 \\ \frac{d}{dx} x^1 & 1\\ \frac{d}{dx} x^2 & 2x \\ \frac{d}{dx} x^3 & 3x^2 \\ \frac{d}{dx} x^4 & 4x^3 \end{array}[/latex]
See the pattern?
[latex]\boxed{\frac{d}{dx} x^n = n x^{n-1}}[/latex]
As we’ll see in the next section, this even works for non-integers. The key is to multiply by the exponent, then decrease the exponent by one.
The next rules say that constant multiples and addition work nicely.
[latex]\boxed{\frac{d}{dx} c \cdot f(x) = c f'(x)}[/latex]
[latex]\boxed{\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)}[/latex]
The key is you can just worry about the derivative of each piece of a sum separately. Constant multiples “come along for the ride”. With only these three rules, you can now take the derivative of any polynomial. Check it out.
Find the following derivatives.
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[latex]\frac{d}{dx} 3x^2[/latex]
To do this one, we use the power rule on the [latex]x^2[/latex] part, and get [latex]2x[/latex]. However, we are also multiplying by [latex]3[/latex], so the answer is multiplied by [latex]3[/latex] as well. Hence the answer is [latex]3(2x) = \boxed{6x}[/latex].
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[latex]\frac{d}{dx} x^3 + x[/latex]
By the power rule, we find [latex]\frac{d}{dx} x^3 = 3x^2[/latex], and [latex]\frac{d}{dx} x[/latex] is [latex]\frac{d}{dx} x^1[/latex] which becomes [latex]1x^0[/latex] by the power rule, which is [latex]1[/latex]. By the addition rule, we have [latex]\frac{d}{dx} x^3 + x = \boxed{3x^2 + 1}[/latex].
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[latex]\frac{d}{dx} 2x^3 + 5[/latex]
You take the derivative of [latex]x^3[/latex] and you have [latex]3x^2[/latex]. Times by [latex]2[/latex], that leaves [latex]6x^2[/latex]. Okay, about the five? It is tempting to leave the five put, but actually [latex]\frac{d}{dx} 5 = 0[/latex]. Why? Well, it’s a constant, so it does not affect the slope. Hence we get [latex]\boxed{6x^2}[/latex].
More power rule examples
Note that the power rule works with fractional and negative exponents as well! Here are some examples.
To apply the power rule in this case, we need to first multiply by the exponent ([latex]{\color{red} 3/2}[/latex]), then subtract one from the exponent [latex]{\color{blue} 3/2 - 1 = 1/2}[/latex]. Then we have
[latex]\frac{d}{dx} x^{3/2} = {\color{red} \frac{3}{2}} x^{{\color{blue} 1/2}} = \boxed{\frac{3}{2} x^{1/2}}[/latex]
In this problem, we just worry about the [latex]x^{-2}[/latex] to start with. We multiply by the exponent [latex]({\color{red} -2})[/latex], and then subtract one from that to get [latex]{\color{blue} -2 - 1 = -3}[/latex]. Then we have
[latex]\frac{d}{dx} 7x^{-2} = 7 ({\color{red} -2} x^{{\color{blue} -3}}) = \boxed{-14 x^{-3}}[/latex]
Sometimes there are hidden fractional or negative exponents. Just remember these rules:
[latex]\boxed{\frac{1}{x^n} = x^{-n}}[/latex] [latex]\boxed{\sqrt[m]{x^n} = \sqrt[m]{x}^n = x^{n/m}}[/latex]
Let’s see some examples.
Fi