| Constant Rule | Examples of Constant, Power, Product and Quotient Rules |
| Power Rule | Derivatives of Trigonometric Functions |
| Product Rule | Higher Order Derivatives |
| Quotient Rule | More Practice |
| List of Rules |
Note that you can use www.wolframalpha.com (or use app on smartphone) to check derivatives by typing in “derivative of x^2(x^2+1)”, for example. Really cool!
I promised you that I’d give you easier way to take derivatives, and the constant, power, product, quotient and basic trigonometry function rules make it much, much easier. Note that there are examples for all these rules here.
Let’s first talk about some notation. When we take the derivative, say of the function $ f\left( x \right)=x+3$, we typically say that we are taking the derivative of $ \boldsymbol {y=f\left( x \right)}$ with respect to $ \boldsymbol {x}$, or whatever the independent variable is (later we’ll take the derivative with respect to more than one variable). When we take the derivative with respect to $ x$, we may see it written the following ways: $ \displaystyle {f}’\left( x \right),\,{y}’,\,\frac{{dy}}{{dx}},\,\frac{{d\left[ {f\left( x \right)} \right]}}{{dx}},\,\frac{d}{{dx}}\left[ {f\left( x \right)} \right]$ (and you may see it differently, such as $ \displaystyle \frac{{df}}{{dx}},\frac{d}{{dx}}y,\,{{D}_{x}}y$). It all basically means the same thing.
Remember that with an algebraic expression that’s a sum or difference (like $ {{x}^{2}}+x+3$), you can separate the expressions and take the derivative of each part (like $ {{x}^{2}}$ and $ x$ and 3) and add (or subtract) them together! This is called the Sum and Difference Rule.
(For really complicated functions that appear to need power, product and/or quotient rules, it may be easier to take the ln (natural logarithm) of each side to differentiate, like we did here in the Exponential and Logarithmic Differentiation section.)
Constant Rule
This is an easy one; whenever we have a constant (a number by itself without a variable), the derivative is just 0. For example, for $ y=3$ , the derivative of that function is just 0. Here is what it looks like in Theorem form:
Power Rule
The way I remember the power rule is take the exponent of a function and move it to the front (to multiply the rest by, including any coefficients), and then take the exponent down a level. In order to use this though, we have to make sure it’s only $ \boldsymbol {x}$ (or whatever the variable is) that’s raised to that exponent; otherwise it gets more complicated (although we can always turn something like $ {{\left( {2x} \right)}^{5}}$ into $ 32{{x}^{5}}$).
As an example of the Power Rule, for $ y=4{{x}^{3}}$, move the 3 in front, and bring the $ x$ cubed down to $ x$ squared: $ \displaystyle \frac{{dy}}{{dx}}=3\cdot 4{{x}^{{3-1}}}=12{{x}^{2}}$. Interestingly enough, the derivative of the volume of a sphere with respect to its radius, $ \displaystyle V=\frac{4}{3}\pi {{r}^{3}}$ is the surface area of a sphere, $ \displaystyle \frac{{dV}}{{dr}}=3\cdot \frac{4}{3}\pi {{r}^{{3-1}}}=4\pi {{r}^{2}}$. Not a coincidence! Here is what it looks like in Theorem form:
Product Rule
The product rule gets a little more complicated, since the derivative of the product of two expressions with variables in them is NOT the product of their derivatives! Only use the product rule if there is some sort of variable in both expressions that you’re multiplying. For example, use it when you have something like $ {{x}^{2}}\left( {x+3} \right)$, but not something like $ \left( {5{{x}^{2}}} \right)\left( 2 \right)$; turn this into $ 10{{x}^{2}}$. Also, if you can, turn exponents on the bottom into negative exponents; for example, $ \displaystyle \frac{5}{{{{x}^{2}}}}=5{{x}^{{-2}}}$. Bottom line: always try to simplify the function first.
Here’s how I like to remember it: the derivative is first times the derivative of the second PLUS second times derivative of the first. Yes, that is a plus (addition) in the middle. Make it into a little song, and it becomes much easier. And notice that typically you have to use the constant and power rules for the individual expressions when you are using the product rule. Here is what it looks like in Theorem form (you may learn it a little differently, like in a different order, but it will come out the same):
Note that if you have a coefficient in front of two factors, you can either lump the coefficient with one of the factors (like the first one), or take it out and multiply the whole derivative later. For example, for $ y=5x{{\left( {x+1} \right)}^{3}}$, the derivative can be obtained this way:
Note how we took out a greatest common factor (GCF) after taking the derivative (like we learned here in the Advanced Factoring section), in order to simplify the expression.
Quotient Rule
First of all, remember that you don’t need to use the quotient rule if there are just numbers (constants) in the denominator – only if there are variables in the denominator!
One thing to remember about the quotient rule is to always start with the bottom, and then it will be easier. I remember it this way: bottom times the derivative of the top minus top times the derivative of the bottom, all over the bottom squared. Note that that the top has a minus in it, not a plus.
Note that if you can separate a quotient into individual terms, it’s best to avoid the quotient rule. For example, if you have the function $ \displaystyle f\left( x \right)=\frac{{{{x}^{3}}+3}}{x}$, it’s best to divide both terms on the top by $ x$ to get $ \displaystyle f\left( x \right)={{x}^{2}}+\frac{3}{x}={{x}^{2}}+3{{x}^{{-1}}}$. (You can’t do this with a function like $ \displaystyle f\left( x \right)=\frac{x}{{{{x}^{3}}+3}}$). And you can always turn a denominator into a numerator using negative exponents, but typically the quotient rule is easier. For example, $ \displaystyle \frac{x}{{{{x}^{3}}+3}}=x{{\left( {{{x}^{3}}+3} \right)}^{{-1}}}$:
Here is what it looks like in Theorem form:
List of Rules
Here are all the rules, with some examples. Do you see how with the product and quotient rules, we may need to use the constant and power rules? Also, when we have a (non-variable) coefficient, it’s typically easier to take it out first before we do the differentiation. Note in all these cases, with what we’ve learned so far with these rules, the coefficient of the $ \boldsymbol {x}$ must be 1 (unless we can take out the coefficient from the whole expression).
More Examples
Here are more examples. Here are more examples. Do you see how important algebra is with calculus?
Here is another example of where we have use the Power Rule twice, since we’re multiplying three factors:
Here are some problems where you have use to the product and quotient rules to find derivatives at certain points using functions, or graphs of functions:
Derivatives of Trigonometric Functions
You basically just have to memorize the derivatives of the Basic Trigonometric Functions. Here they are:
Here are a few examples; note that the sum and difference rules (and all the other rules) apply for trig derivatives, too:
Higher Order Derivatives
We can actually take the derivative of a function more than once; we’ll see this here in the Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change Section. We’ll see that the acceleration function is the derivative of the velocity function, which is the derivative of the position function. (In this case, we would take the derivative twice of the position function to get the acceleration function.)
The second derivative (and third derivative, and so on) is what we call a higher order derivative, and the notation looks like the following:
(Don’t let all this scare you; you’ll usually just be using one or two types of notation, and it will be pretty obvious). And with polynomials, if we keep taking derivatives, we’ll eventually end up with 0; for example, for $ y=5{{x}^{4}}-3{{x}^{2}}+2x+3$, $ \begin{array}{c}{y}’=20{{x}^{3}}-6x+2,\,\,\,\,\,{y}”=60{{x}^{2}}-6\,\,\\{y}”’=120x;\,\,\,\,\,\,{{y}^{{\left( 4 \right)}}}=120;\,\,\,\,\,\,{{y}^{{\left( 5 \right)}}}=0\end{array}$.
Here are more problems; note that in the first case, we have what we call “indestructible derivatives” with the sin and cos, since we can keep taking the derivative forever, and the functions never go away (you won’t end up with 0, as in the previous example with a polynomial).
Understand these problems, and practice, practice, practice!
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On to Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change – you’re ready!