$$\frac{d}{dx} [x^n] = nx^{n-1}$$

Derivative of $$\sqrt{u}$$

$$\frac{d}{dx} [\sqrt{u}] = \frac{1}{2\sqrt{u}} \frac{du}{dx}$$

Derivative of $$\frac{1}{x}$$

$$\frac{d}{dx} \left[\frac{1}{x}\right] = -\frac{1}{x^2}$$

$$\frac{d}{dx} [f(x)g(x)] = f$$

$$\frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f$$

$$\frac{d}{dx} [\sin u] = \cos u \frac{du}{dx}$$

Derivative of $$\cos u$$

$$\frac{d}{dx} [\cos u] = -\sin u \frac{du}{dx}$$

$$\frac{d}{dx} [\tan u] = \sec^2 u \frac{du}{dx}$$

Derivative of $$\cot u$$

$$\frac{d}{dx} [\cot u] = -\csc^2 u \frac{du}{dx}$$

$$\frac{d}{dx} [\sec u] = \sec u \tan u \frac{du}{dx}$$

Derivative of $$\csc u$$

$$\frac{d}{dx} [\csc u] = -\csc u \cot u \frac{du}{dx}$$

$$\frac{d}{dx} [\ln u] = \frac{1}{u} \frac{du}{dx}$$

Derivative of $$\log_a u$$

$$\frac{d}{dx} [\log_a u] = \frac{1}{u \ln a} \frac{du}{dx}$$

$$\frac{d}{dx} [e^u] = e^u \frac{du}{dx}$$

$$\frac{d}{dx} [a^u] = a^u \ln a \frac{du}{dx}$$

Derivative of arcsin u

$$\frac{d}{dx} [\arcsin u] = \frac{1}{\sqrt{1-u^2}} \frac{du}{dx}$$

Derivative of arccos u

$$\frac{d}{dx} [\arccos u] = -\frac{1}{\sqrt{1-u^2}} \frac{du}{dx}$$

Derivative of arctan u

$$\frac{d}{dx} [\arctan u] = \frac{1}{1+u^2} \frac{du}{dx}$$

Derivative of arccot u

$$\frac{d}{dx} [\text{arc cot} u] = -\frac{1}{1+u^2} \frac{du}{dx}$$

Definition of a definite integral

$$\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{k=1}^n f(x_k) \cdot (\Delta x_k)$$

Definition of a Critical Number

Let f be defined at c. If f'(c) = 0 or if f' is undefined at c, then c is a critical number of f.

First Derivative Test - Relative Minimum

If f'(x) changes from negative to positive at c, then (c, f(c)) is a relative minimum of f.

First Derivative Test - Relative Maximum

If f'(x) changes from positive to negative at c, then (c, f(c)) is a relative maximum of f.

Definition of Concavity

Let f be differentiable on an open interval I. The graph of f is concave upward on I if f' is increasing on the interval and concave downward on I if f' is decreasing on the interval.

Test for Concavity - Concave Upward

If f''(x) > 0 for all x in I, then the graph of f is concave upward in I.

Definition of an Inflection Point

A function f has an inflection point at (c, f(c)) if f''(c) = 0 or f''(c) does not exist and f'' changes sign at x = c.

Second Fundamental Theorem of Calculus

$$\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$$

Chain Rule Version of 2nd Fundamental Theorem of Calculus

$$\frac{d}{dx} \int_{a}^{g(x)} f(t) dt = f(g(x)) \cdot g$$

Average value of f(x) on [a, b]

$$\frac{1}{b-a}\cdot\int_a^bf(x)dx$$

Rate In/Rate Out formula

$$\int_a^b(\text{rate of change})dt$$ = amount of change from t = a to t = b

L'Hospital's Rule conditions

If f and g are differentiable and g'(x) ≠ 0 near x = a, and $$\lim_{x \to a} f(x) = 0$$ and $$\lim_{x \to a} g(x) = 0$$ or $$\lim_{x \to a} f(x) = \pm \infty$$ and $$\lim_{x \to a} g(x) = \pm \infty$$ Then $$\lim_{x\rightarrow a}\frac{f\left(x\right)}{g\left(x\right)}=\lim_{x\rightarrow a}\frac{f$$

L'Hospital's Rule application

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f$$

Integration by Parts: Indefinite Formula

$$\int_{ }^{ }u\cdot dv=uv-\int_{ }^{ }v\cdot du$$

Integration by Parts: Definite Formula

$$\int_{a}^{b} u dv = [uv]_{a}^{b} - \int_{a}^{b} v du$$

Order for Choosing "u" in Integration by Parts

L - Logarithms, I - Inverse Trig Functions, P - Polynomials, E - Exponential Functions, T - Trig Functions

Length of a Curve (x-axis)

$$L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} dx$$

Calculus Derivatives and Theorems Flashcards

Student preview

78 questions

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1.

FLASHCARD QUESTION

Front

Definition of the derivative

Back

2.

FLASHCARD QUESTION

Front

Alternative form of the derivative

Back

3.

FLASHCARD QUESTION

Front

Definition of continuity

Back

4.

FLASHCARD QUESTION

Front

Mean Value Theorem

Back

5.

FLASHCARD QUESTION

Front

Intermediate Value Theorem

Back

If f is continuous on [a, b] and k is any number between f(a) and f(b), then there is at least one number c between a and b such that f(c) = k.

6.

FLASHCARD QUESTION

Front

Derivative of xⁿ

Back

7.

FLASHCARD QUESTION

Front

Back

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