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Derivatives & Differentiation Rules cheat sheet - grade 11-12

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Calculus Grade 11-12

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.

TopicsDerivativesFunctionsTrigonometryGraphsCalculusStudy Guide
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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)f'(x) gives the slope of the tangent line to y=f(x)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 yy.

Key Facts

  • The derivative definition is f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} when the limit exists.
  • The power rule is ddxxn=nxn−1\frac{d}{dx}x^n=nx^{n-1} for any real number nn where the derivative is defined.
  • Constant and sum rules are ddx[cf(x)]=cf′(x)\frac{d}{dx}[cf(x)]=cf'(x) and ddx[f(x)+g(x)]=f′(x)+g′(x)\frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x).
  • The product rule is ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x).
  • The quotient rule is ddx[f(x)g(x)]=g(x)f′(x)−f(x)g′(x)[g(x)]2\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)≠0g(x)\ne 0.
  • The chain rule is ddxf(g(x))=f′(g(x))g′(x)\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 ddxsin⁡x=cos⁡x\frac{d}{dx}\sin x=\cos x, ddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x=-\sin x, and ddxtan⁡x=sec⁡2x\frac{d}{dx}\tan x=\sec^2 x.
  • Common exponential and logarithmic derivatives include ddxex=ex\frac{d}{dx}e^x=e^x, ddxax=axln⁡a\frac{d}{dx}a^x=a^x\ln a, and ddxln⁡x=1x\frac{d}{dx}\ln x=\frac{1}{x} for x>0x>0.

Vocabulary

Derivative
The derivative f′(x)f'(x) is the instantaneous rate of change of f(x)f(x) with respect to xx.
Differentiable
A function is differentiable at x=ax=a if f′(a)f'(a) exists, which means the graph has a well-defined tangent slope there.
Tangent Line
A tangent line at x=ax=a is the line with slope f′(a)f'(a) passing through (a,f(a))(a,f(a)), written y−f(a)=f′(a)(x−a)y-f(a)=f'(a)(x-a).
Chain Rule
The chain rule differentiates a composite function using ddxf(g(x))=f′(g(x))g′(x)\frac{d}{dx}f(g(x))=f'(g(x))g'(x).
Implicit Differentiation
Implicit differentiation finds dydx\frac{dy}{dx} from an equation involving both xx and yy by treating yy as a function of xx.
Critical Number
A critical number is a value x=cx=c in the domain of ff where f′(c)=0f'(c)=0 or f′(c)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, ddx(3x+1)5=5(3x+1)4⋅3\frac{d}{dx}(3x+1)^5=5(3x+1)^4\cdot 3, not just 5(3x+1)45(3x+1)^4.
  • Using the product rule as f′g′f'g' instead of f′g+fg′f'g+fg', which ignores how both factors change. The correct rule is ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x)\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 g(x)f′(x)−f(x)g′(x)[g(x)]2\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}.
  • Dropping negative signs in trigonometric derivatives, which leads to wrong slopes. Remember ddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x=-\sin x and ddxcot⁡x=−csc⁡2x\frac{d}{dx}\cot x=-\csc^2 x.
  • Treating yy as a constant during implicit differentiation, which misses the factor dydx\frac{dy}{dx}. For example, ddxy2=2ydydx\frac{d}{dx}y^2=2y\frac{dy}{dx}.

Practice Questions

  1. 1 Differentiate f(x)=4x5−3x2+7x−9f(x)=4x^5-3x^2+7x-9.
  2. 2 Find dydx\frac{dy}{dx} for y=(2x3−5x)4y=(2x^3-5x)^4.
  3. 3 Find the equation of the tangent line to f(x)=x2+3xf(x)=x^2+3x at x=2x=2.
  4. 4 Explain why the chain rule is needed to differentiate sin⁡(5x2)\sin(5x^2) instead of using only ddxsin⁡x=cos⁡x\frac{d}{dx}\sin x=\cos x.

Understanding Derivatives & Differentiation Rules

Differentiation rules are not a list to apply at random. Each rule matches the structure of an expression. A useful habit is to identify the last operation performed on the variable.

If a quantity is put inside another function, the change must pass through both layers. For example, the area of a circle depends on its radius.

If the radius changes over time, the area changes because of the radius and because area responds differently at different radii. This layered effect explains why an inner rate of change must be included when differentiating a composite expression.

Before differentiating, scan the expression for grouping symbols, powers, multiplication, division, and functions inside functions. An expression such as x squared times sine x has two changing factors, so it needs the product rule. Sine of x squared has one function inside another, so it needs the chain rule.

These look similar at first, but their structures are different. Simplifying can make the work shorter.

Expanding a simple product or rewriting a fraction with negative powers may help. Do not simplify in a way that hides important restrictions, especially when a denominator could equal zero.

A derivative is not guaranteed at every input. Corners, cusps, jumps, and some vertical tangents can cause trouble. The graph of the absolute value of x has a sharp corner at zero, so there is no single slope there.

Domain restrictions matter too. A logarithm requires a positive input in the usual real number setting. A quotient is undefined wherever its denominator is zero.

Trigonometric derivative formulas use radians, not degrees. This is a common source of wrong answers because a calculator may be in degree mode while calculus formulas assume radian measure.

Implicit differentiation is useful when x and y are connected by an equation that is awkward or impossible to solve neatly for y. Treat y as a quantity that changes when x changes. Every time a term containing y is differentiated, include the rate dy over dx through the chain rule.

After finding that rate, substitute the coordinates of the required point. The result can be used to build a tangent line or to estimate nearby values. In real situations, derivatives appear as velocity, population growth, changing temperature, electrical current, and marginal cost.

Check answers by looking at signs, units, and the graph. A positive derivative should match increasing behavior nearby, while a negative derivative should match decreasing behavior.