Domanda del libro di testo
For problems 49–52 use implicit differentiation to find dy/dx at the given point P.
51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Inverse Trigonometric Functions
Scelta multipla
Find the derivative of the given function.
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Guida verificata passo dopo passo1
Identify the function for which you need to find the derivative: \( f(t) = \csc^{-1}(2t+7) \).
Recall the derivative formula for the inverse cosecant function: \( \frac{d}{dx}[\csc^{-1}(x)] = -\frac{1}{|x|\sqrt{x^2-1}} \).
Apply the chain rule to differentiate \( f(t) = \csc^{-1}(2t+7) \). The chain rule states that \( \frac{d}{dt}[\csc^{-1}(u)] = \frac{d}{du}[\csc^{-1}(u)] \cdot \frac{du}{dt} \), where \( u = 2t+7 \).
Differentiate \( u = 2t+7 \) with respect to \( t \), which gives \( \frac{du}{dt} = 2 \).
Substitute \( u = 2t+7 \) and \( \frac{du}{dt} = 2 \) into the chain rule expression: \( \frac{d}{dt}[\csc^{-1}(2t+7)] = -\frac{2}{|2t+7|\sqrt{(2t+7)^2-1}} \).
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