130. Use the identity arccot(u)=π/2 - arctan(u) to derive the formula for the derivative of arccot(u) in Table 7.4 from the formula for the derivative of arctan(u).

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Recall the given identity: \(\arccot(u) = \frac{\pi}{2} - \arctan(u)\).

Differentiate both sides of the equation with respect to \(x\). Since \(\frac{\pi}{2}\) is a constant, its derivative is zero, so we have: \(\frac{d}{dx} [\arccot(u)] = \frac{d}{dx} \left[ \frac{\pi}{2} - \arctan(u) \right] = - \frac{d}{dx} [\arctan(u)]\).

Use the chain rule to differentiate \(\arctan(u)\): \(\frac{d}{dx} [\arctan(u)] = \frac{1}{1 + u^2} \cdot \frac{du}{dx}\).

Substitute this result back into the derivative of \(\arccot(u)\): \(\frac{d}{dx} [\arccot(u)] = - \frac{1}{1 + u^2} \cdot \frac{du}{dx}\).

Thus, the derivative formula for \(\arccot(u)\) is derived as \(\frac{d}{dx} [\arccot(u)] = - \frac{1}{1 + u^2} \cdot \frac{du}{dx}\), matching the formula in Table 7.4.

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주요 개념

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Inverse Trigonometric Functions

Inverse trigonometric functions, like arctan and arccot, are the inverses of the tangent and cotangent functions, respectively. They return the angle whose tangent or cotangent is a given number. Understanding their definitions and domains is essential for manipulating and differentiating these functions.

Derivative of arctan(u)

The derivative of arctan(u) with respect to x is given by (1 / (1 + u^2)) times the derivative of u. This formula is fundamental because it provides a direct way to differentiate arctan functions, which can then be used to find derivatives of related inverse trig functions through identities.

Using Identities to Derive Derivatives

By expressing one inverse trig function in terms of another using identities, such as arccot(u) = π/2 - arctan(u), we can apply known derivatives and the chain rule to find new derivatives. This method simplifies the process by leveraging existing formulas rather than differentiating from first principles.