Table of Contents
Differentiation rules are practical formulas that let you find derivatives quickly, without going back to the limit definition every time. In this chapter we collect the most commonly used rules and see how to apply them.
We will assume you already know what a derivative is (as a limit and as an instantaneous rate of change) from the chapter on the definition of the derivative. Here we focus on how to compute derivatives efficiently.
1. Basic function derivatives
These are derivatives of simple, standard functions. You use them as building blocks.
- Constant function
If $c$ is a constant, then
$$
\frac{d}{dx}(c) = 0.
$$
- Power function
For any real number $n$,
$$
\frac{d}{dx}\bigl(x^n\bigr) = n x^{n-1},
$$
provided $x^n$ is defined (e.g. for $n$ an integer this holds for all $x$ where $x^n$ makes sense).
Examples:
- $\dfrac{d}{dx}(x^2) = 2x$
- $\dfrac{d}{dx}(x^5) = 5x^4$
- $\dfrac{d}{dx}(x^{-2}) = -2x^{-3}$
- $\dfrac{d}{dx}(\sqrt{x}) = \dfrac{d}{dx}(x^{1/2}) = \dfrac{1}{2}x^{-1/2}$
- Exponential functions
For the natural exponential:
$$
\frac{d}{dx}\bigl(e^x\bigr) = e^x.
$$
For a general base $a>0$, $a\neq 1$:
$$
\frac{d}{dx}\bigl(a^x\bigr) = a^x \ln(a).
$$
- Logarithmic functions
For the natural logarithm:
$$
\frac{d}{dx}\bigl(\ln x\bigr) = \frac{1}{x}, \quad x>0.
$$
For logarithm base $a>0$, $a\neq 1$:
$$
\frac{d}{dx}\bigl(\log_a x\bigr) = \frac{1}{x \ln(a)}, \quad x>0.
$$
- Trigonometric functions
For angles measured in radians:
$$
\frac{d}{dx}(\sin x) = \cos x,
$$
$$
\frac{d}{dx}(\cos x) = -\sin x,
$$
$$
\frac{d}{dx}(\tan x) = \sec^2 x \quad\text{(where defined)}.
$$
- Inverse trigonometric functions
On their natural domains,
$$
\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}},
$$
$$
\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}},
$$
$$
\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}.
$$
These basic derivatives are always combined with the rules below, especially the chain rule, when the inside of the function is not simply $x$.
2. Linearity: sum and constant multiple rules
Differentiation is linear. This is the foundation for differentiating polynomials and many other expressions.
- Sum and difference rule
If $f$ and $g$ are differentiable, then
$$
\frac{d}{dx}\bigl(f(x) + g(x)\bigr) = f'(x) + g'(x),
$$
$$
\frac{d}{dx}\bigl(f(x) - g(x)\bigr) = f'(x) - g'(x).
$$
- Constant multiple rule
If $c$ is a constant and $f$ is differentiable, then
$$
\frac{d}{dx}\bigl(c\,f(x)\bigr) = c\,f'(x).
$$
These two rules justify:
- Differentiating each term of a polynomial separately.
- Pulling constant coefficients outside the derivative.
Example:
$$
\frac{d}{dx}\bigl(3x^4 - 5x + 7\bigr)
= 3\cdot 4x^3 - 5\cdot 1 + 0
= 12x^3 - 5.
$$
3. Product rule
When two functions are multiplied, you cannot simply multiply their derivatives. Instead, use:
Product rule
If $f$ and $g$ are differentiable, then
$$
\frac{d}{dx}\bigl(f(x)g(x)\bigr)
= f'(x)g(x) + f(x)g'(x).
$$
A common way to remember: “first times derivative of second plus second times derivative of first.”
Example:
Differentiate $h(x) = x^2\sin x$.
- $f(x) = x^2,\; f'(x) = 2x$
- $g(x) = \sin x,\; g'(x) = \cos x$
Then
$$
h'(x) = 2x\sin x + x^2\cos x.
$$
The product rule extends to products of more than two factors, applied repeatedly. For example, for three functions $u,v,w$:
$$
\frac{d}{dx}(uvw) = u'vw + uv'w + uvw'.
$$
4. Quotient rule
When one function is divided by another, a similar but distinct rule applies.
Quotient rule
If $f$ and $g$ are differentiable and $g(x)\neq 0$, then
$$
\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)
= \frac{f'(x)g(x) - f(x)g'(x)}{\bigl(g(x)\bigr)^2}.
$$
A common memory aid: “(low d‑high minus high d‑low) over (low) squared”, where “high” = numerator, “low” = denominator.
Example:
Differentiate
$$
y = \frac{x^2 + 1}{x}.
$$
Here, $f(x) = x^2 + 1,\; f'(x) = 2x$; $g(x) = x,\; g'(x) = 1$.
Then
$$
y' = \frac{(2x)\cdot x - (x^2 + 1)\cdot 1}{x^2}
= \frac{2x^2 - x^2 - 1}{x^2}
= \frac{x^2 - 1}{x^2}.
$$
Sometimes it is simpler to rewrite before differentiating: in this example,
$$
y = \frac{x^2 + 1}{x} = x + x^{-1},
$$
so
$$
y' = 1 - x^{-2} = 1 - \frac{1}{x^2}.
$$
Both methods agree after simplification. Choosing between quotient rule and algebraic simplification is a practical skill: often you can avoid the quotient rule by rewriting.
5. Chain rule
Many functions are compositions: one function applied inside another. The chain rule handles these.
Chain rule
If $y = f(g(x))$, where $f$ and $g$ are differentiable, then
$$
\frac{dy}{dx} = f'(g(x)) \cdot g'(x).
$$
A useful way to say it: differentiate the outer function, keep the inside the same, then multiply by the derivative of the inside.
5.1 Recognizing compositions
Think of $x$ passing through several “layers”:
- Inner layer: $u = g(x)$
- Outer layer: $y = f(u)$
Then
$$
\frac{dy}{dx} = \frac{dy}{du}\cdot \frac{du}{dx}.
$$
Once you get used to it, you often apply the chain rule directly without explicitly naming $u$, but using $u$ is very helpful while learning.
5.2 Typical patterns
- Power of a function
For $y = [g(x)]^n$,
$$
\frac{dy}{dx} = n[g(x)]^{n-1}\cdot g'(x).
$$
Example:
$$
y = (3x^2 + 1)^4, \quad
y' = 4(3x^2 + 1)^3 \cdot (6x) = 24x(3x^2 + 1)^3.
$$
- Exponential with nontrivial exponent
For $y = e^{g(x)}$,
$$
\frac{dy}{dx} = e^{g(x)}\cdot g'(x).
$$
Example:
$$
y = e^{x^2}, \quad y' = e^{x^2}\cdot 2x = 2xe^{x^2}.
$$
- Logarithm of a function
For $y = \ln(g(x))$ (with $g(x) > 0$),
$$
\frac{dy}{dx} = \frac{g'(x)}{g(x)}.
$$
Example:
$$
y = \ln(5x^3 - 1), \quad
y' = \frac{15x^2}{5x^3 - 1}.
$$
- Trig functions with a function inside
For $y = \sin(g(x))$,
$$
\frac{dy}{dx} = \cos(g(x))\cdot g'(x).
$$
Example:
$$
y = \sin(2x^3), \quad
y' = \cos(2x^3)\cdot 6x^2 = 6x^2\cos(2x^3).
$$
Chain rule compositions can nest multiple times; in that case, you apply the chain rule repeatedly, layer by layer.
6. Higher-order derivatives
Once you know how to compute first derivatives, you can differentiate again to get higher-order derivatives.
- The second derivative of $y=f(x)$ is
$$
f''(x) = \frac{d}{dx}\bigl(f'(x)\bigr) = \frac{d^2y}{dx^2}.
$$ - The third derivative is
$$
f'''(x) = \frac{d}{dx}\bigl(f''(x)\bigr).
$$ - For $n\ge 1$, the $n$‑th derivative is denoted $f^{(n)}(x)$.
You compute these by reapplying the rules from this chapter to the derivative you just found. Higher derivatives are central in later topics like curve sketching and differential equations, but here the focus is just on how to obtain them.
Example:
Let $f(x) = x^4$.
- First derivative: $f'(x) = 4x^3$.
- Second derivative: $f''(x) = 12x^2$.
- Third derivative: $f'''(x) = 24x$.
- Fourth derivative: $f^{(4)}(x) = 24$.
- Fifth derivative and beyond: $0$.
7. Derivatives of combinations of standard functions
Most practical derivatives use several rules together. The general strategy:
- Break the function into recognizable pieces.
- Apply sum and constant multiple rules to separate terms.
- Decide whether you need product, quotient, or chain rules (or a combination).
- Apply basic function derivatives and simplify.
To illustrate the interplay, here are a few examples that mix the rules.
- Example: $y = x^2 e^x$
This is a product, so use the product rule.
- $f(x) = x^2,\; f'(x) = 2x$
- $g(x) = e^x,\; g'(x) = e^x$
Then
$$
y' = f'(x)g(x) + f(x)g'(x)
= 2x e^x + x^2 e^x
= e^x(2x + x^2).
$$
- Example: $y = \dfrac{\sin x}{x^2}$
Use the quotient rule.
- $f(x) = \sin x,\; f'(x) = \cos x$
- $g(x) = x^2,\; g'(x) = 2x$
Then
$$
y' = \frac{\cos x\cdot x^2 - \sin x\cdot 2x}{x^4}
= \frac{x^2\cos x - 2x\sin x}{x^4}.
$$
You can simplify by factoring out $x$ if desired.
- Example: $y = (x^2 + 1)\ln x$
Product rule plus basic derivatives.
- $f(x) = x^2 + 1,\; f'(x) = 2x$
- $g(x) = \ln x,\; g'(x) = 1/x$
Then
$$
y' = 2x\ln x + (x^2 + 1)\cdot \frac{1}{x}
= 2x\ln x + x + \frac{1}{x}.
$$
- Example: $y = \ln(\sin x)$
This is a composition: outer function $\ln(\cdot)$, inner function $\sin x$.
- Outer derivative: $\dfrac{d}{du}(\ln u) = \dfrac{1}{u}$
- Inner derivative: $\dfrac{d}{dx}(\sin x) = \cos x$
By chain rule,
$$
y' = \frac{1}{\sin x}\cdot \cos x = \cot x
$$
(when $\sin x\neq 0$).
- Example: $y = \sqrt{1 + e^{3x}}$
Rewrite as a power to see the structure:
$$
y = \bigl(1 + e^{3x}\bigr)^{1/2}.
$$
Outer: $u^{1/2}$, inner: $1 + e^{3x}$.
- Outer derivative: $\dfrac{d}{du}\bigl(u^{1/2}\bigr) = \dfrac{1}{2}u^{-1/2}$
- Inner derivative: $\dfrac{d}{dx}(1 + e^{3x}) = 3e^{3x}$
Chain rule:
$$
y' = \frac{1}{2}\bigl(1 + e^{3x}\bigr)^{-1/2}\cdot 3e^{3x}
= \frac{3e^{3x}}{2\sqrt{1 + e^{3x}}}.
$$
- Example: repeated chain and product: $y = x^2 e^{x^3}$
- Product of $x^2$ and $e^{x^3}$, so product rule.
- Inside of the exponential is $x^3$, so chain rule.
Let $f(x)=x^2,\; f'(x)=2x$; $g(x)=e^{x^3}$.
For $g(x)$:
- Outer: $e^u$ with derivative $e^u$
- Inner: $u=x^3$ with derivative $3x^2$
So
$$
g'(x) = e^{x^3}\cdot 3x^2.
$$
Now product rule:
$$
y' = 2x\cdot e^{x^3} + x^2\cdot 3x^2 e^{x^3}
= 2x e^{x^3} + 3x^4 e^{x^3}
= e^{x^3}(2x + 3x^4).
$$
In all such examples, the rules from this chapter are the tools you apply in sequence to compute the derivative.
8. Using notation fluently
You will see and use several equivalent notations for derivatives:
- $f'(x)$
- $\dfrac{dy}{dx}$ when $y=f(x)$
- $Df$ or simply $Df(x)$
- For higher derivatives: $f''(x)$, $f^{(n)}(x)$, $\dfrac{d^n y}{dx^n}$
When applying rules like the chain rule, it is often convenient to think in terms of $\dfrac{dy}{dx}$ and intermediate variables like $u=g(x)$, but in other contexts $f'(x)$ may be more compact. Being comfortable with all these notations helps when reading and writing derivatives.
The rules in this chapter are the practical toolkit of differential calculus. In later chapters on applications of derivatives, you will use them repeatedly to analyze functions, optimize quantities, and model change.