Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 3 - Derivatives

Problema 3.5.34a

Capitolo 3, Problema 3.5.34a

Find y⁽⁴⁾ = d⁴y/dx⁴ if:
a. y = −2 sin x

Guida verificata passo dopo passo

Start by identifying the function y = -2 sin(x). We need to find the fourth derivative of this function with respect to x.

Recall that the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). These derivatives alternate as you continue differentiating.

Calculate the first derivative: y' = d/dx [-2 sin(x)] = -2 cos(x).

Calculate the second derivative: y'' = d/dx [-2 cos(x)] = 2 sin(x).

Calculate the third derivative: y''' = d/dx [2 sin(x)] = 2 cos(x). Finally, calculate the fourth derivative: y⁽⁴⁾ = d/dx [2 cos(x)] = -2 sin(x).

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Higher-Order Derivatives

Higher-order derivatives refer to the derivatives of a function taken multiple times. The first derivative represents the rate of change of the function, the second derivative indicates the curvature, and so on. In this case, finding the fourth derivative means applying the differentiation process four times to the function y = -2 sin x.

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Higher Order Derivatives

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. The sine function, in particular, oscillates between -1 and 1 and has specific derivatives: the derivative of sin x is cos x, and the derivative of cos x is -sin x. Understanding these properties is essential for differentiating functions involving trigonometric terms.

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Percorso guidato

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Introduction to Trigonometric Functions

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. While not directly applied in this specific question, it is crucial for more complex functions involving trigonometric identities.

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Intro to the Chain Rule

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Interpreting Derivative Values

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Differentiability and Continuity on an Interval

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