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[h]SL 5.3 Derivative of f(x)
[q] Polynomial Derivatives
As we said, you won’t need to use the formula using limits, but we will use it here to derive a rule for derivatives of polynomials.
E.G.: Find the derivative of f(x) = x² – 4x + 2:
We will use the “first principles” formula this one time:
\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) – f(x)}{h} = \lim_{h \to 0} \frac{(x² + 2xh + h² – 4x – 4h + 2) – (x² – 4x + 2)}{h} = \lim_{h \to 0} \frac{2xh + h² – 4h}{h} = 2x – 4
\]
As h → 0, this simplifies to 2x – 4.
[q]
This same process can be applied to a general polynomial \(ax^n + bx^m + …\), and what we get is:
\[
f'(x) = anx^{n-1} + bmx^{m-1} + …
\]
This can be described as taking each term and multiplying it by the exponent, then subtracting one from the exponent.
[a]
Formula Booklet: In the formula booklet, we are only given the formula:
\[
f(x) = x^n \quad \Rightarrow \quad f'(x) = nx^{n-1}
\]
So you need to know that the constant does not affect this rule, and that we can seamlessly add terms using this rule for each term.
[q]
Differentiation: This is simply another term used for the process of finding the derivative.
Examples:
E.G. : Differentiate f(x) = 4x⁷:
If f(x) = 4x⁷, then f'(x) = 4(7)x⁶ = 28x⁶.
E.G. : Find the derivative of y = 3x³ – 12x² + 8x + 1:
We use the “anx” rule again. So:
\[
\frac{dy}{dx} = 3(3)x² – 12(2)x + 8(1) = 9x² – 24x + 8
\]
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