Mathematics, 03.11.2019 21:31 student679

What is the third derivative of tan x?

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Answer from: IamHowToBasic

$\displaystyle y''' = 2 \sec^2 (x) \bigg( 2\tan^2 (x) + \sec^2 (x) \bigg)$

General Formulas and Concepts:

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Multiplied Constant]: $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]: $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]: $\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

Derivative Rule [Chain Rule]: $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle y = \tan x$

Step 2: Differentiate

Trigonometric Differentiation: $\displaystyle y' = \sec^2 (x)$ Basic Power Rule [Derivative Rule - Chain Rule]: $\displaystyle y'' = 2 \sec (x) \cdot \frac{d}{dx}[\sec (x)]$ Trigonometric Differentiation: $\displaystyle y'' = 2 \sec (x) \cdot \sec (x) \tan (x)$ Simplify: $\displaystyle y'' = 2 \sec^2 (x) \tan (x)$ Derivative Rule [Product Rule]: $\displaystyle y''' = \frac{d}{dx}[2 \sec^2 (x)] \tan (x) + 2 \sec^2 (x) \frac{d}{dx}[\tan (x)]$ Rewrite [Derivative Property - Multiplied Constant]: $\displaystyle y''' = 2 \frac{d}{dx}[\sec^2 (x)] \tan (x) + 2 \sec^2 (x) \frac{d}{dx}[\tan (x)]$ Trigonometric Differentiation: $\displaystyle y''' = 2 \frac{d}{dx}[\sec^2 (x)] \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)$ Basic Power Rule [Derivative Rule - Chain Rule]: $\displaystyle y''' = 2 \big( 2 \sec (x) \big) \frac{d}{dx}[\sec (x)] \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)$ Trigonometric Differentiation: $\displaystyle y''' = 2 \big( 2 \sec (x) \big) \big( \sec (x) \tan (x) \big) \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)$ Simplify: $\displaystyle y''' = 4 \sec^2 (x) \tan^2 (x) + 2 \sec^4 (x)$ Factor: $\displaystyle y''' = 2 \sec^2 (x) \bigg( 2\tan^2 (x) + \sec^2 (x) \bigg)$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

Answer from: Quest

this shows a reflection over the x-axis and is translated one unit down and 6 units to the right (x + 6, y - 1)

hope this !

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Find the composition of transformations that map abcd to ehgf. !

Answer from: Quest

Do the math. she originally had 50 dollars on her gift card and she had 12 dollars left on her gift card after she finished her shopping spree. if you subtract 50 minus 12 you get thirty eight dollars as your answer. so therefore, ciara spent 38 dollars total o

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