Mathematics: Determine the second derivative of y=cosxsinx

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Determine the second derivative of y=cosxsinx

Ответ:

naseersaad

31.12.2021

Mathematics

$\displaystyle \frac{dy}{dx} = -4 \sin(x) \cos(x)$

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

$\displaystyle y = \cos(x) \sin(x)$

Step 2: Differentiate

Derivative Rule [Product Rule]: $\displaystyle y' = \frac{d}{dx}[\cos(x)] \sin(x) + \cos(x) \frac{d}{dx}[\sin(x)]$ Trigonometric Differentiation: $\displaystyle y' = -\sin^2(x) + \cos^2(x)$ Derivative Property [Addition/Subtraction]: $\displaystyle y' = \frac{d}{dx}[-\sin^2(x)] + \frac{d}{dx}[\cos^2(x)]$ Rewrite [Derivative Property - Multiplied Constant]: $\displaystyle y' = -\frac{d}{dx}[\sin^2(x)] + \frac{d}{dx}[\cos^2(x)]$ Basic Power Rule [Derivative Rule - Chain Rule]: $\displaystyle y' = -2 \sin(x) \frac{d}{dx}[\sin(x)] + 2 \cos(x) \frac{d}{dx}[\cos(x)]$ Trigonometric Differentiation: $\displaystyle y' = -2 \sin(x) \cos(x) - 2 \cos(x) \sin(x)$ Simplify: $\displaystyle y' = -4 \sin(x) \cos(x)$

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

Unit: Differentiation

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Determine the second derivative of y=cosxsinx

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