Calculus Solver Derivative rules
Seven rules take any expression apart. A second list gives you the derivatives you memorise rather than derive. Everything below has one worked example next to it, because a formula on its own is the part nobody has trouble with.
The seven rules
These are structural. They do not tell you the derivative of anything in particular, they tell you how to break a complicated expression into simpler ones.
| Rule | Formula | Example |
|---|---|---|
| Constant | d/dx [c] = 0 | d/dx [7] = 0 |
| Power | d/dx [xⁿ] = n·xⁿ⁻¹ | d/dx [x⁵] = 5x⁴ |
| Constant multiple | d/dx [c·f] = c·f′ | d/dx [4x³] = 12x² |
| Sum and difference | d/dx [f ± g] = f′ ± g′ | d/dx [x² + sin x] = 2x + cos x |
| Product | d/dx [f·g] = f′g + fg′ | d/dx [x²·eˣ] = 2x·eˣ + x²·eˣ |
| Quotient | d/dx [f/g] = (f′g − fg′) / g² | d/dx [x / (x+1)] = 1 / (x+1)² |
| Chain | d/dx [f(g(x))] = f′(g(x))·g′(x) | d/dx [sin(3x)] = 3cos(3x) |
Common derivatives
This is the memorisation half. Everything here can be derived from the definition, and in a first course you will do exactly that once and then never again.
| f(x) | f′(x) | Notes |
|---|---|---|
| c | 0 | |
| xⁿ | n·xⁿ⁻¹ | Works for negative and fractional n |
| √x | 1 / (2√x) | The power rule with n = ½ |
| 1/x | −1 / x² | The power rule with n = −1 |
| eˣ | eˣ | The only function that is its own derivative |
| aˣ | aˣ·ln a | |
| ln x | 1 / x | For x > 0 |
| logₐ x | 1 / (x·ln a) | |
| sin x | cos x | Radians, not degrees |
| cos x | −sin x | Radians, not degrees |
| tan x | sec²x | |
| cot x | −csc²x | |
| sec x | sec x·tan x | |
| csc x | −csc x·cot x | |
| arcsin x | 1 / √(1 − x²) | |
| arccos x | −1 / √(1 − x²) | Same as arcsin with the sign flipped |
| arctan x | 1 / (1 + x²) |
Print this section or screenshot it. Between the two tables it is the whole cheat sheet, and there is nothing else in a first calculus course that you need on paper.
Which rule to use
Knowing the formulas is the easy half. Reading an expression and seeing which one applies is where people actually get stuck. Go down this list in order.
Terms added or subtracted?
Split it and differentiate each piece on its own. Always do this first, before you go looking for a harder rule. Most problems that look terrifying are three easy problems written next to each other.
A number in front?
Pull it out and forget about it until the end. The constant multiple rule is the one people skip mentioning and use constantly.
Two functions multiplied?
Product rule. The test for whether it really is a product: can you name two separate things that each depend on x? In x²·eˣ you can. In x²ˣ you cannot, that is one function.
One function divided by another?
Quotient rule works, but rewriting it as f·g⁻¹ and using the product and chain rules together is usually less error-prone, because the quotient rule's minus sign is the most commonly dropped symbol in calculus.
Something inside something else?
Chain rule. sin(3x²) is sin of a thing, and the thing is 3x². Differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.
All of the above at once?
Work from the outermost structure inwards. In x²·sin(3x) the top level is a product, so start with the product rule, and the chain rule turns up later when you differentiate sin(3x).
Four mistakes worth knowing about
Every one of these has a five second check attached to it. Use the check, not your memory.
The product rule is not f′·g′
Test it on x·x, which is x². The real derivative is 2x. Multiplying the derivatives gives 1·1 = 1. The rule is f′g + fg′, and it has two terms because both factors are moving at once.
The chain rule is the inner derivative
d/dx [sin(3x)] is 3cos(3x), not cos(3x). The factor of 3 is the entire content of the rule. If your answer to a chain rule problem has no new factor in it, you have not applied the rule.
The quotient rule order
The numerator is f′g − fg′, in that order, and swapping the terms flips the sign of the whole answer. Check yourself on 1/x: f = 1, g = x gives (0·x − 1·1) / x² = −1 / x². If you got +1 / x², the terms are backwards.
Trig derivatives assume radians
d/dx [sin x] = cos x is only true when x is in radians. In degrees there is a factor of π/180 in front, which is exactly why nobody does calculus in degrees.
Questions
What are the seven derivative rules?
The constant rule, the power rule, the constant multiple rule, the sum and difference rule, the product rule, the quotient rule and the chain rule. Those seven tell you how to take apart any expression. Separately from them you need the derivatives of the standard functions, which are the second table above, and those are memorised rather than derived every time.
What is the derivative of 3x?
It is 3. The constant multiple rule pulls the 3 out, the power rule turns x into 1, and 3 times 1 is 3. More generally the derivative of any straight line is its slope, which is the whole reason the derivative is worth defining.
Do I actually need to memorise the quotient rule?
No. Every quotient f/g can be written f·g⁻¹ and handled with the product rule and the chain rule, which you need anyway. Plenty of people go through calculus without the quotient rule and lose nothing. If your course requires it on an exam, memorise it, but check your sign on 1/x every time.
Why is the derivative of eˣ equal to eˣ?
For any base, d/dx [aˣ] = aˣ·ln a. The number e is defined as the base where that extra factor equals 1, because ln e = 1. So e is not a magic number that happens to behave this way. It is the number picked precisely so that it would.
Does the chain rule stack for three layers?
Yes, and it keeps going for as many layers as you have. d/dx [sin(cos(3x))] is cos(cos(3x)) times −sin(3x) times 3. Work outside in and multiply everything you collect on the way down. Miss one layer and the answer is wrong by exactly that factor.
Stuck on one in particular
Point the camera at the problem and the app names the rule at each line instead of handing you an answer with no working. Then ask the tutor why that rule and not another one.
