Derivatives & Differentiation Rules Cheat Sheet

A printable reference covering derivative definitions, power, product, quotient, chain, trigonometric, exponential, logarithmic, implicit differentiation, and tangent lines for grades 11-12.

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Derivatives measure how a function changes at an instant, making them essential for slopes, rates, motion, optimization, and graph analysis. This cheat sheet organizes the main differentiation rules students use in first calculus courses. It is designed as a fast binder reference for Grade 11-12 practice, homework, and review. Knowing these rules helps students move from long limit calculations to efficient symbolic differentiation. The core idea is that the derivative $f'(x)$ gives the slope of the tangent line to $y=f(x)$ at a point. Basic rules such as the power rule, product rule, quotient rule, and chain rule handle most algebraic functions. Special formulas cover trigonometric, exponential, and logarithmic functions. Implicit differentiation and tangent line equations connect derivatives to curves that are not solved directly for $y$ .

Key Facts

The derivative definition is $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ when the limit exists.
The power rule is $\frac{d}{dx}x^n=nx^{n-1}$ for any real number $n$ where the derivative is defined.
Constant and sum rules are $\frac{d}{dx}[cf(x)]=cf'(x)$ and $\frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x)$ .
The product rule is $\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$ .
The quotient rule is $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}$ , where $g(x)\ne 0$ .
The chain rule is $\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$ , so the outer derivative is multiplied by the inner derivative.
Common trigonometric derivatives include $\frac{d}{dx}\sin x=\cos x$ , $\frac{d}{dx}\cos x=-\sin x$ , and $\frac{d}{dx}\tan x=\sec^2 x$ .
Common exponential and logarithmic derivatives include $\frac{d}{dx}e^x=e^x$ , $\frac{d}{dx}a^x=a^x\ln a$ , and $\frac{d}{dx}\ln x=\frac{1}{x}$ for $x>0$ .

Vocabulary

Derivative: The derivative $f'(x)$ is the instantaneous rate of change of $f(x)$ with respect to $x$ .
Differentiable: A function is differentiable at $x=a$ if $f'(a)$ exists, which means the graph has a well-defined tangent slope there.
Tangent Line: A tangent line at $x=a$ is the line with slope $f'(a)$ passing through $(a,f(a))$ , written $y-f(a)=f'(a)(x-a)$ .
Chain Rule: The chain rule differentiates a composite function using $\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$ .
Implicit Differentiation: Implicit differentiation finds $\frac{dy}{dx}$ from an equation involving both $x$ and $y$ by treating $y$ as a function of $x$ .
Critical Number: A critical number is a value $x=c$ in the domain of $f$ where $f'(c)=0$ or $f'(c)$ does not exist.

Common Mistakes to Avoid

Forgetting the inner derivative in the chain rule, which makes derivatives of composites incomplete. For example, $\frac{d}{dx}(3x+1)^5=5(3x+1)^4\cdot 3$ , not just $5(3x+1)^4$ .
Using the product rule as $f'g'$ instead of $f'g+fg'$ , which ignores how both factors change. The correct rule is $\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$ .
Reversing the quotient rule numerator, which changes the sign of the answer. The correct order is $\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}$ .
Dropping negative signs in trigonometric derivatives, which leads to wrong slopes. Remember $\frac{d}{dx}\cos x=-\sin x$ and $\frac{d}{dx}\cot x=-\csc^2 x$ .
Treating $y$ as a constant during implicit differentiation, which misses the factor $\frac{dy}{dx}$ . For example, $\frac{d}{dx}y^2=2y\frac{dy}{dx}$ .

Practice Questions

1 Differentiate $f(x)=4x^5-3x^2+7x-9$ .
2 Find $\frac{dy}{dx}$ for $y=(2x^3-5x)^4$ .
3 Find the equation of the tangent line to $f(x)=x^2+3x$ at $x=2$ .
4 Explain why the chain rule is needed to differentiate $\sin(5x^2)$ instead of using only $\frac{d}{dx}\sin x=\cos x$ .