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Calculus I Formula Sheet - Explained

Derivatives & Limits - With Full Explanations

4 Pages Beginner MATH 140, MATH 151, CALC 1 $6.99

LIMITS: What Are We Approaching?

Limits are about predicting where a function is HEADED, even if it never actually gets there. Think of it like watching someone walk toward a wall - you can tell where they're going even before they arrive.

Limit Definition
$$\lim_{x \to a} f(x) = L$$

As $x$ gets closer and closer to $a$, the function value $f(x)$ gets closer and closer to $L$. It's like zooming in on a graph at a specific point and asking "what y-value are we approaching?" The function doesn't need to actually reach $L$ at $x=a$ - we only care about the approach.

x - The input variable moving toward $a$
a - The x-value we're approaching
f(x) - The function we're evaluating
L - The limit value
! The limit and the actual function value can be DIFFERENT! A function can have a limit of 5 at $x=2$ but actually equal 7 at $x=2$ (or be undefined there).
Example: $\lim_{x \to 2} \frac{x^2-4}{x-2} = \lim_{x \to 2} \frac{(x+2)(x-2)}{x-2} = \lim_{x \to 2} (x+2) = 4$. Even though the original is undefined at $x=2$, the limit is 4.
Special Trig Limit
$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

As $x$ gets tiny, $\sin x$ and $x$ become nearly identical, so their ratio approaches 1. This is one of the most important limits in calculus!

! $x$ MUST BE IN RADIANS, not degrees! This limit doesn't work with degrees.
Example: $\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{x \to 0} \frac{3\sin(3x)}{3x} = 3 \cdot 1 = 3$
L'Hopital's Rule
$$\text{If } \frac{0}{0} \text{ or } \frac{\infty}{\infty}: \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$

When you get $\frac{0}{0}$ or $\frac{\infty}{\infty}$, take derivatives of top and bottom separately (not the quotient rule!).

TIP You can apply L'Hopital's Rule multiple times if you keep getting indeterminate forms.

DERIVATIVE SHORTCUT RULES

These rules let you find derivatives in seconds instead of using the limit definition every time.

Power Rule
$$\frac{d}{dx}[x^n] = n \cdot x^{n-1}$$

The most-used rule in calculus. Bring the exponent down as a coefficient, then reduce the exponent by 1. Works for ANY real exponent - positive, negative, or fractional.

n - Any real number exponent
x - The variable
! Don't forget: $\frac{d}{dx}[x] = 1$, not $x$. And $\frac{d}{dx}[x^0] = 0$ since $x^0 = 1$ (a constant).
Examples: $\frac{d}{dx}[x^5] = 5x^4$ and $\frac{d}{dx}[\sqrt{x}] = \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$
Product Rule
$$\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$

"First times derivative of second, plus second times derivative of first." The derivative of a product is NOT just the product of the derivatives!

! $(fg)' \neq f'g'$ - This is a common mistake! You MUST use the product rule.
Example: $\frac{d}{dx}[x \sin x] = (1)(\sin x) + (x)(\cos x) = \sin x + x\cos x$
Chain Rule
$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Derivative of outer times derivative of inner. When you have a function inside another function, work from outside in.

TIP Think "peel the onion" - differentiate the outer layer, keep the inner intact, then multiply by the derivative of the inner.
Example: $\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x = 2x\cos(x^2)$

COMMON DERIVATIVES TABLE

Memorize these - they show up everywhere in calculus.

Exponential & Logarithm
$$\frac{d}{dx}[e^x] = e^x \qquad \frac{d}{dx}[\ln x] = \frac{1}{x}$$

$e^x$ is the ONLY function that equals its own derivative. Natural log gives you $\frac{1}{x}$ - one of the cleanest derivatives.

! Power rule does NOT apply to $e^x$! It's NOT $xe^{x-1}$. And $\frac{d}{dx}[\ln x] = \frac{1}{x}$, not $\frac{1}{\ln x}$.
Trig Derivatives
$$\frac{d}{dx}[\sin x] = \cos x \qquad \frac{d}{dx}[\cos x] = -\sin x \qquad \frac{d}{dx}[\tan x] = \sec^2 x$$

Notice the pattern: the "co-" functions (cosine, cotangent, cosecant) have negative derivatives.

TIP Don't forget chain rule! $\frac{d}{dx}[\sin(2x)] = 2\cos(2x)$, not just $\cos(2x)$.

APPLICATIONS OF DERIVATIVES

Tangent Line
$$y - y_1 = m(x - x_1)$$
Critical Points
$$f'(x) = 0$$

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Calculus I Formula Sheet - Standard

Quick Reference - All Formulas, No Fluff

2 Pages Beginner MATH 140, MATH 151, CALC 1 $3.99

LIMIT LAWS

Sum/Difference
$\lim_{x \to a}[f(x) \pm g(x)] = L \pm M$
Product
$\lim_{x \to a}[f(x) \cdot g(x)] = L \cdot M$
Quotient
$\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{L}{M}$
Sine Limit
$\lim_{x \to 0}\frac{\sin x}{x} = 1$
Euler's Number
$\lim_{x \to \infty}\left(1 + \frac{1}{x}\right)^x = e$

DERIVATIVE RULES

Power Rule
$\frac{d}{dx}[x^n] = nx^{n-1}$
Product Rule
$\frac{d}{dx}[fg] = f'g + fg'$
Quotient Rule
$\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$
Chain Rule
$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

TRIG DERIVATIVES

Sine
$\frac{d}{dx}[\sin x] = \cos x$
Cosine
$\frac{d}{dx}[\cos x] = -\sin x$
Tangent
$\frac{d}{dx}[\tan x] = \sec^2 x$

EXPONENTIAL & LOGARITHM

Exponential
$\frac{d}{dx}[e^x] = e^x$
Natural Log
$\frac{d}{dx}[\ln x] = \frac{1}{x}$
General Exponential
$\frac{d}{dx}[a^x] = a^x \ln a$
General Log
$\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$

INVERSE TRIG & APPLICATIONS

Arcsin
$\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}$
Arctan
$\frac{d}{dx}[\arctan x] = \frac{1}{1+x^2}$

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