Differentiation rules

Differentiation is the process of finding the derivative of a function.

These rules help you find derivatives without having to use the limit definition every time. Think of them as shortcuts.

Derivative Rules to Memorize

Brief Examples

Power Rule

Rule: $\frac{d}{dx}{x}^n=n{x}^{n-1}$
Mnemonic: Bring the exponent down as a coefficient and subtract 1 from the exponent.
Example: $\frac{d}{dx}{x}^3=3x^2$

Constant Rule

Rule: $\frac{d}{dx}c=0$
Mnemonic: Constants don’t change, so their derivative is zero.
Example: $\frac{d}{dx}5=0$

Sum/Difference Rule

Rule: $\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)$
Mnemonic: Differentiate each term separately and keep the operation.
Example: $\frac{d}{dx}(x^2+3x)=2x+3$

Product Rule

Rule: $\frac{d}{dx}[f(x)g(x)]=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$
Mnemonic: First times derivative of second, plus second times derivative of first.
Example: $\frac{d}{dx}(x\sin x)=\sin x+x\cos x$

Quotient Rule

Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$
Mnemonic: Derivative of top times bottom minus top times derivative of bottom, over bottom squared.
Example: $\frac{d}{dx}\left(\frac{x^2}{x+1}\right)=\frac{x^2+2x}{(x+1{)}^2}$

Chain Rule

Rule: $\frac{d}{dx}f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x)$
Mnemonic: Derivative of the outer function (evaluated at the inner) times derivative of the inner function.
Example: $\frac{d}{dx}(x^2+1{)}^3=6x(x^2+1{)}^2$

1. Power Rule

For functions like $x^n$.

Rule: $\frac{d}{dx}{x}^n=n{x}^{n-1}$

Brief Example: $\frac{d}{dx}{x}^3=3x^2$

Detailed Examples:

Example: Find the derivative of $f(x)=x^3$

• Step 1: Identify the exponent. Here, $n=3$.
• Step 2: Multiply the exponent by $x$: $3\cdot x=3x$
• Step 3: Subtract 1 from the exponent: $3-1=2$
• Step 4: Put it together: $3x^2$

So, the derivative is $3x^2$.

Another Example: $f(x)=x^5$

• $n=5$
• $5\cdot x=5x$
• $5-1=4$
• Derivative: $5x^4$

2. Constant Rule

For constant functions.

Rule: $\frac{d}{dx}c=0$

Brief Example: $\frac{d}{dx}5=0$

Detailed Examples:

Example: $f(x)=7$

• This is a constant, so derivative is 0.

Example: $f(x)=-3$

• Still a constant, derivative is 0.

3. Sum/Difference Rule

For sums or differences of functions.

Rule: $\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)$

Brief Example: $\frac{d}{dx}(x^2+3x)=2x+3$

Detailed Examples:

Example: $f(x)=x^2+3x+1$

• Break it down: $x^2+3x+1$
• Derivative of $x^2$ (Power Rule): $2x$
• Derivative of $3x$ (Power Rule): $3$
• Derivative of $1$ (Constant Rule): $0$
• Add them up: $2x+3+0=2x+3$

Example: $f(x)=x^3-2x$

• $x^3$ derivative: $3x^2$
• $2x$ derivative: $2$
• Subtract: $3x^2-2$

4. Product Rule

For products of functions.

Rule: $\frac{d}{dx}[f(x)g(x)]=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$

Brief Example: $\frac{d}{dx}(x\sin x)=\sin x+x\cos x$

Detailed Example:

Example: $f(x)=x\cdot \sin (x)$

• Let $u=x$, $v=\sin (x)$
• $u^{\prime }=1$ (derivative of $x$)
• $v^{\prime }=\cos (x)$ (derivative of $\sin (x)$)
• Product Rule: $(1\cdot \sin (x))+(x\cdot \cos (x))=\sin (x)+x\cos (x)$

Step-by-step:

• First part: derivative of $x$ is 1, times $\sin (x)$: $1\cdot \sin (x)=\sin (x)$
• Second part: $x$ times derivative of $\sin (x)$: $x\cdot \cos (x)$
• Add them: $\sin (x)+x\cos (x)$

5. Quotient Rule

For quotients of functions.

Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$

Brief Example: $\frac{d}{dx}\left(\frac{x^2}{x+1}\right)=\frac{x^2+2x}{(x+1{)}^2}$

Detailed Example:

Example: $f(x)=\frac{x^2}{x+1}$

• Let $u=x^2$, $v=x+1$
• $u^{\prime }=2x$
• $v^{\prime }=1$
• Numerator: $(2x)(x+1)-(x^2)(1)=2x(x+1)-x^2=2x^2+2x-x^2=x^2+2x$
• Denominator: $(x+1{)}^2$
• So, derivative: $\frac{x^2+2x}{(x+1{)}^2}$

Step-by-step:

• Top part: $2x\cdot (x+1)=2x^2+2x$
• Bottom part: $x^2\cdot 1=x^2$
• Subtract: $2x^2+2x-x^2=x^2+2x$
• Divide by $(x+1{)}^2$

6. Chain Rule

For composite functions.

Rule: $\frac{d}{dx}f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x)$

Brief Example: $\frac{d}{dx}(x^2+1{)}^3=6x(x^2+1{)}^2$

Detailed Example:

Example: $f(x)=(x^2+1{)}^3$

• Let $u=x^2+1$, so $f=u^3$
• $f^{\prime }$ with respect to $u$: $3u^2$
• $g^{\prime }(x)=$ derivative of inside: $2x$
• So, $3u^2\cdot 2x=3(x^2+1{)}^2\cdot 2x=6x(x^2+1{)}^2$

Step-by-step:

• Inside function: $x^2+1$, its derivative: $2x$
• Outside: ${}^3$, derivative of $u^3$ is $3u^2$
• Multiply: $3u^2\cdot 2x$
• Substitute back: $3(x^2+1{)}^2\cdot 2x=6x(x^2+1{)}^2$

Example: Differentiating $\frac{-4x^2-6x+2}{x^2}$

This example demonstrates how to differentiate a complex rational function by breaking it down using the differentiation rules.

Step 1: Simplify the Expression

First, divide each term in the numerator by the denominator:
$\frac{-4x^2-6x+2}{x^2}=-4-\frac{6}{x}+\frac{2}{x^2}$

Step 2: Differentiate Each Term

• Constant Rule for -4: The derivative of a constant is 0.
• Power Rule for $-\frac{6}{x}=-6x^{-1}$: Bring the exponent down and subtract 1: $-6\cdot (-1)x^{-2}=6x^{-2}=\frac{6}{x^2}$
• Power Rule for $\frac{2}{x^2}=2x^{-2}$: Bring the exponent down and subtract 1: $2\cdot (-2)x^{-3}=-4x^{-3}=-\frac{4}{x^3}$

Step 3: Sum the Derivatives (Sum/Difference Rule)

Add them up: $0+\frac{6}{x^2}-\frac{4}{x^3}=\frac{6}{x^2}-\frac{4}{x^3}$

Alternative: Using Quotient Rule Directly

Let $f(x)=-4x^2-6x+2$, $g(x)=x^2$
$f^{\prime }(x)=-8x-6$ (Power Rule and Sum/Difference Rule)
$g^{\prime }(x)=2x$ (Power Rule)

Quotient Rule: $\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}=\frac{(-8x-6)x^2-(-4x^2-6x+2)(2x)}{x^4}$

Compute numerator:
$(-8x-6)x^2=-8x^3-6x^2$
$-(-4x^2-6x+2)(2x)=-(-8x^3-12x^2+4x)=8x^3+12x^2-4x$

Total numerator: $-8x^3-6x^2+8x^3+12x^2-4x=6x^2-4x$

Derivative: $\frac{6x^2-4x}{x^4}=\frac{6}{x^2}-\frac{4}{x^3}$

This example shows the Power Rule applied to each term, Constant Rule for constants, Sum/Difference Rule for combining, and Quotient Rule as an alternative method.