The constant rule

This is simple. is a horizontal line with a slope of zero, and thus its derivative is also zero. So, for any number c, if , then . Or you can write . End of story.

The power rule

Say . To find its derivative, take the power, 5, bring it in front of the x, and reduce the power by 1 (in this example, the power becomes a 4). That gives you . Piece o' cake.

In Chapter 4, I differentiate with the difference quotient. It takes eight lines of math to do it. Instead of all that, just use the power rule: Bring the 2 in front, reduce the power by 1, which leaves you with a power of 1 that you can drop (because a power of 1 does nothing). Thus,

Once you know the nifty shortcuts like this one, you’ll rarely use the difference quotient. So why bother learning all that difficult difference quotient stuff? Well, the difference quotient is included in every calculus book and course because it gives you a fuller, richer understanding of the calculus and its foundations — think of it as a mathematical character builder. Or because math teachers are sadists. You be the judge.

The power rule works for positive, negative, and fraction powers.

Make sure you remember how to do the last function. It’s the simplest function, yet the easiest problem to miss. You can just memorize that the derivative of x is 1. Or do it like this: Rewrite as and then apply the power rule: Bring the 1 in front and reduce the power by 1 to zero, giving you . Because equals 1, you’ve got .

You can differentiate radical functions by rewriting them as power functions and then using the power rule. For example, if , rewrite it as and use the power rule. You can also use the power rule to differentiate functions like . Rewrite this as and use the power rule.

The constant multiple rule

What if the function you’re differentiating begins with a coefficient? Makes no difference. A coefficient has no effect on differentiation. Just ignore it and differentiate according to the appropriate rule. The coefficient stays where it is until the final step when you simplify your answer by multiplying by the coefficient.

Differentiate .

Solution: You know by the power rule that the derivative of is , so the derivative of is . The 4 just sits there doing nothing. Then, as a final step, you simplify: equals . So .

Differentiate .

Solution: This is a line of the form with , so the slope is 5 and thus the derivative is 5:. (It’s important to think graphically like this sometimes.) But you can also solve it with the power rule.; so .

In a nutshell, the constant multiple rule takes a function like , differentiates the stuff — that’s — while the 10 just stays put. So, if , then .

Don’t forget that (~3.14) and e (~2.72) are numbers, not variables, so they behave like ordinary numbers. Constants in problems, like c and k, also behave like ordinary numbers.

Thus, if — this works exactly like differentiating . And because is just a number, if then — this works exactly like differentiating . You’ll also see problems containing constants like c and k. Be sure to treat them like regular numbers. For example, the derivative of (where k is a constant) is 5, not .

The sum and difference rules

When you want the derivative of a sum of terms, take the derivative of each term separately.

What’s

Just use the power rule for each of the first four terms and the constant rule for the final term. Thus, .

If you’ve got a difference (that’s subtraction) instead of a sum, it makes no difference. You still differentiate each term separately. Thus, if , then .

Differentiating trig functions

I have the high honor and distinct privilege of introducing you to the derivatives of the six trig functions.

Make sure you memorize the derivatives for sine and cosine — they’re a snap. You should learn the other four as well, but if you’re afraid that this knowledge will crowd out the date of the Battle of Hastings (1066), you can use the following nifty mnemonic device I made up.

Imagine you’re taking a test and can’t remember these four derivatives. You lean over to the guy next to you and whisper, “Psst, hey what’s the derivative of cscx?” Now, take the last three letters of psst (sst) — those are the initial letters of sec, sec, tan. Write these three down, and below them write their cofunctions: csc, csc, cot. Put a negative sign on the csc in the middle. Finally, add arrows like in the following diagram.

The sec on the left has an arrow pointing to sec tan — so the derivative of secx is secx tanx. The tan on the right has an arrow pointing to sec sec, so the derivative of tanx is . The bottom row works the same except both derivatives are negative. Believe it or not, this trick is easy to remember.

Exponential and logarithmic functions

If you can’t memorize the next rule, hang up your calculator.

That’s right, the derivative of is itself! This is a special function. and its multiples, like , are the only functions that are their own derivatives.

If the base is a number other than e, you have to tweak the derivative by multiplying it by the natural log of the base:

• If , then .
• If , then .

Here’s the derivative of the natural log:

If the log base is a number other than e, you tweak this derivative — as with exponential functions — except you divide by the natural log of the base instead of multiplying:

The product and quotient rules

The Product Rule (for the product (duh) of two functions):

So, for

The Quotient Rule (bet you can guess what this is for):

Most calculus books give this rule in a slightly different form that’s harder to remember. And some give a “mnemonic” involving the words lodeehi and hideelo or hodeehi and hideeho, which is easy to get mixed up — great, thanks a lot.

Memorize the quotient rule as I’ve written it. You’ll remember what goes in the denominator — no one ever forgets it. The trick is knowing the order of the terms in the numerator. Think of it like this: You’re doing a derivative, so the first thing you do is to take a derivative. The natural place to begin is at the top of the fraction. So the quotient rule begins with the derivative of the top. Remember that, and the rest of the numerator is almost automatic.

Here’s the derivative of :

The chain rule

The chain rule is by far the trickiest derivative rule, but it’s not really that bad if you carefully focus on a few important points. Let’s begin by differentiating . You use the chain rule here because you’ve got a composite function, that’s one function inside another function (the square root function).

Here’s one way to quickly recognize a composite function. is not a composite function because the argument of the square root — the thing you take the square root of — is simply x. Whenever the argument of a function is anything other than a plain old x, you’ve got a composite function. Be careful to distinguish a composite function from something like , which is the product of two functions, and sin x, each of which does have just an x as its argument.

Okay, so you’ve got this composite function, . Here’s how to differentiate it with the chain rule.

1. You start with the outside function, , and differentiate that, IGNORING what’s inside. To make sure you ignore the inside, temporarily replace the inside function with the word stuff.

   So you’ve got . Okay, now differentiate the same way you’d differentiate . Because is the same as , the power rule gives you . So for this problem, you begin with .

2. Multiply the result from Step 1 by the derivative of the inside function, .
   

   Take a good look at this. All basic chain rule problems follow this format. You do the derivative rule for the outside function, ignoring the inside stuff, then multiply that by the derivative of the stuff.

3. Differentiate the inside stuff.

   The inside stuff in this problem is , and its derivative is by the power rule.

4. Now put the real stuff and its derivative back where they belong.
   

   Or, if you’ve got something against negative powers, . Or, if you’ve got something against fraction powers, .

Let’s differentiate another composite function: .

1. The outside function is the sine function, so you start there, taking the derivative of sine and ignoring the inside stuff, . The derivative of sin x is cos x, so for this problem, you begin with
   
2. Multiply the derivative of the outside function by the derivative of the stuff.
   
3. The stuff in this problem is , so is 2x. When you plug these terms back in, you get
   

Sometimes figuring out which function is inside which can be tricky — especially when a function is inside another and then both of them are inside a third (you can have four or more nested functions, but three is probably the most you’ll see).

Rewrite the composite function with a set of parentheses around each inside function, and rewrite trig functions like with the power outside a set of parentheses: .

For example — this is tough — differentiate . First, rewrite the cubed sine function: . Now it’s easy to see the order in which the functions are nested. The innermost function is in the innermost parentheses — that’s . Next, the sine function is in the next set of parentheses — that’s . Last, the cubing function is outside everything — that’s . (Did you notice that the stuff in is different from the stuff in ? I admit this is quite unmathematical, but don’t sweat it. I’m just using the term stuff to refer to whatever is inside any function.) Now that you know the order of the functions, you differentiate from outside in.

1. The outermost function is and its derivative is given by the power rule.
   
2. As with all chain rule problems, you multiply that by .
   
3. Now put the stuff, , back where it belongs.
   

4. Use the chain rule again.

   You can’t finish this quickly by just taking a simple derivative because you have to differentiate another composite function, Just treat as if it were the original problem and take its derivative. The derivative of sin x is cos x, so the derivative of begins with . Multiply that by . Thus, the derivative of is

   
5. The stuff for this step is and its derivative is . Plug those things back in.
   
6. Now that you’ve got the derivative of , plug this result into the result from Step 3, giving you the whole enchilada.
   

You can save some time by not switching to the word stuff and then switching back. But some people like to use the technique because it forces them to leave the stuff alone during each step of a problem. That’s the critical point.

Make sure you … DON’T TOUCH THE STUFF.

Don’t change the inside function while differentiating the outside one. Say you want to differentiate . The argument of this natural logarithm function is . Don’t touch it during the first step of the solution, which is to use the natural log rule: , which says to put the argument of the natural log function in the denominator under the 1. So, after the first step in differentiating , you’ve got .

Finish by multiplying that by the derivative of , which is .

Final answer: .

Another way to avoid chain rule mistakes is to remember that you never use more than one derivative rule at a time.

In the previous example, , you first use the natural log rule, then, as a separate step, you use the power rule to differentiate . At no point in any chain rule problem do you use both rules at the same time and write something like .

The Chain Rule (for differentiating a composite function):

• If ,
• then .

Or, equivalently,

• If ,
• then (notice how the dus cancel).

Finally, differentiate . This one has a new twist — it involves the chain rule and the product rule. How do you begin?

If you’re not sure where to begin differentiating a complex expression, imagine plugging a number into x and then evaluating the expression on your calculator one step at a time. Your last computation tells you the first thing to do.

Say you plug the number 5 into the xs in . You evaluate — that’s 100; then, after getting , you do , which is about . Finally, you multiply 100 by . Because your last computation is multiplication, your first step in differentiating is to use the product rule. (Had your last computation been something like , you’d begin with the chain rule.) So for , start with the product rule:

Now finish by taking the derivative of with the power rule and the derivative of with the chain rule: