Derivative Rules

You came to class carrying a conjecture. The Derivative ended by asking what $f(x)=x^{17}$ would do, and in the Visual Patterns warm-up your group lined up the evidence: $x^2$ gives $2x$, $x^3$ gives $3x^2$, $x^4$ gives $4x^3$. The exponent hops down front; the new exponent is one less. Nobody handed you the power rule — you caught it in the act.

$\text{If }f(x)=x^n,\text{ then }{f}^{\prime }(x)=n{x}^{n-1}$

The definition proved it for each case your group checked, one limit at a time. The rule is the pattern with a certificate.

The toolbox

Rule	Statement	What it lets you do
Power	$(x^n{)}^{\prime }=n{x}^{n-1}$	any power, even $x^{1/2}$
Constant	$(c{)}^{\prime }=0$	flat graphs have slope zero
Constant multiple	$(cf{)}^{\prime }=c{f}^{\prime }$	scale factors ride along
Sum and difference	$(f\pm g{)}^{\prime }=f^{\prime }\pm g^{\prime }$	one term at a time

The last three are the reasonable rules — the ones the water-barrel argument in class made obvious. If $f(t)$ and $g(t)$ are litres in two barrels, then $f^{\prime }(t)+g^{\prime }(t)$ is how fast the total is rising, and of course that equals $(f+g{)}^{\prime }(t)$. Together the four rules dismantle any polynomial: $f(x)=2x^3+3x^2$ surrenders term by term to $f^{\prime }(x)=6x^2+6x$, no limits required.

One caution worth writing in your Math Journal: the power rule also handles rational exponents, so $f(x)=\sqrt{x}=x^{1/2}$ gives $f^{\prime }(x)=\frac{1}{2}{x}^{-1/2}$ — the rule is broader than the natural numbers you conjectured it from, and that generosity gets verified, not assumed.

Products refuse to cooperate

At the boards your group tested the tempting guess — that the derivative of a product is the product of the derivatives — and it failed on the very first example. It is a good mistake; treat it as data. The rule that actually works keeps both factors in play:

$(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$

Each factor takes a turn changing while the other holds still. For $f(x)=(3x+2)(2x^2-1)$:

$f^{\prime }(x)=3(2x^2-1)+(3x+2)(4x)=18x^2+8x-3$

Expand first instead and differentiate $6x^3+4x^2-3x-2$ — the same $18x^2+8x-3$ appears. Two roads, one answer: that agreement is the verification the course keeps asking for.

Shortcuts remember; you understand

The rules are shortcuts, and shortcuts are earned. Every one of them compresses a limit computation you have done by hand, and when a rule ever feels like magic, the definition is one page away. Where slopes are wanted on demand — a tangent line here, a rate there — the toolbox delivers them in seconds, and Derivative Rules Practice builds that speed honestly, without worshipping it. Functions hiding inside other functions need one more idea: The Chain Rule.

Curriculum connection

A3.1

verify the power rule for functions of the form $f(x)=x^n$, where $n$ is a natural number [e.g., by determining the equations of the derivatives of the functions $f(x)=x$, $f(x)=x^2$, $f(x)=x^3$, and $f(x)=x^4$ algebraically using ${\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$ and graphically using slopes of tangents]

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A3.2

verify the constant, constant multiple, sum, and difference rules graphically and numerically [e.g., by using the function $g(x)=kf(x)$ and comparing the graphs of $g^{\prime }(x)$ and $k{f}^{\prime }(x)$; by using a table of values to verify that $f^{\prime }(x)+g^{\prime }(x)=(f+g{)}^{\prime }(x)$, given $f(x)=x$ and $g(x)=3x$], and read and interpret proofs involving ${\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$ of the constant, constant multiple, sum, and difference rules (student reproduction of the development of the general case is not required) Sample problem: The amounts of water flowing into two barrels are represented by the functions $f(t)$ and $g(t)$. Explain what $f^{\prime }(t)$, $g^{\prime }(t)$, $f^{\prime }(t)+g^{\prime }(t)$, and $(f+g{)}^{\prime }(t)$ represent. Explain how you can use this context to verify the sum rule, $f^{\prime }(t)+g^{\prime }(t)=(f+g{)}^{\prime }(t)$.

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A3.3

determine algebraically the derivatives of polynomial functions, and use these derivatives to determine the instantaneous rate of change at a point and to determine point(s) at which a given rate of change occurs Sample problem: Determine algebraically the derivative of $f(x)=2x^3+3x^2$ and the point(s) at which the slope of the tangent is 36.

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A3.4

verify that the power rule applies to functions of the form $f(x)=x^n$, where $n$ is a rational number [e.g., by comparing values of the slopes of tangents to the function $f(x)=x^{1/2}$ with values of the derivative function determined using the power rule], and verify algebraically the chain rule using monomial functions [e.g., by determining the same derivative for $f(x)=(5x^3{)}^{1/3}$ by using the chain rule and by differentiating the simplified form, $f(x)=5^{1/3}x$] and the product rule using polynomial functions [e.g., by determining the same derivative for $f(x)=(3x+2)(2x^2-1)$ by using the product rule and by differentiating the expanded form $f(x)=6x^3+4x^2-3x-2$] Sample problem: Verify the chain rule by using the product rule to look for patterns in the derivatives of $f(x)=x^2+1$, $f(x)=(x^2+1{)}^2$, $f(x)=(x^2+1{)}^3$, and $f(x)=(x^2+1{)}^4$.

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