Calculus Reference
Derivative Rules
Derivative rules turn common expressions into slopes without rebuilding the difference quotient every time.
Solved Examples
Jump directly to a worked example by problem type.
Core Rules
| Rule | Statement | Use |
|---|---|---|
| Constant | $\frac{d}{dx}[c]=0$ | A constant has no rate of change. |
| Power | $\frac{d}{dx}[x^n]=nx^{n-1}$ | Move the exponent down, then reduce it by $1$. |
| Constant multiple | $(cf)'=cf'$ | Keep the coefficient and differentiate the function. |
| Sum or difference | $(f\pm g)'=f'\pm g'$ | Differentiate each term separately. |
| Product | $(fg)'=f'g+fg'$ | Use when two variable expressions are multiplied. |
| Quotient | $\left(\frac fg\right)'=\frac{f'g-fg'}{g^2}$ | Use when one variable expression is divided by another. |
| Chain | $\frac d{dx}f(g(x))=f'(g(x))g'(x)$ | Differentiate the outside, then multiply by the inside derivative. |
Derivative of a Power Expression
Power Rule
Find the derivative of the expression $f(x)=4x^3-2x+7$.
Solution Steps
- Differentiate each power term using the power rule.
- Differentiate the linear and constant terms, then state the derivative: $f'(x)=12x^2-2$.
$$f'(x)=4(3x^2)-2(1)+0$$$$f'(x)=12x^2-2$$
Derivative of a Sum
Differentiate Term by Term
Find the derivative of the expression $g(x)=x^4+3x^2-5x+1$.
Solution Steps
- Differentiate each term in the polynomial separately.
- Apply the power rule, combine like terms, and state the derivative: $g'(x)=4x^3+6x-5$.
$$g'(x)=4x^3+3(2x)-5$$$$g'(x)=4x^3+6x-5$$
Derivative of a Product
Two Factors
Find the derivative of the expression $h(x)=x^2(x+3)$.
Solution Steps
- Identify the two factors $f=x^2$ and $g=x+3$.
- Apply the product rule, expand, and state the derivative: $h'(x)=3x^2+6x$.
$$h'(x)=(2x)(x+3)+x^2(1)$$$$h'(x)=3x^2+6x$$
Derivative of a Composite Expression
Chain Rule
Find the derivative of the expression $p(x)=(2x+1)^4$.
Solution Steps
- Identify the outside power and inside expression $2x+1$.
- Apply the chain rule, multiply by the inside derivative $2$, and state the derivative: $p'(x)=8(2x+1)^3$.
$$p'(x)=4(2x+1)^3(2)$$$$p'(x)=8(2x+1)^3$$