Question

Prove the formula for \( (d / d x)\left(\cos ^{-1}(x)\right) \) by the same method as for \( (d / d x)\left(\sin ^{-1}(x)\right) \). Let \( y=\cos ^{-1}(x) \). Then \( \cos (y)= \) \( \square \) and \( 0 \leq y \leq \pi \Rightarrow-\sin (y) \frac{d y}{d x}=1 \Rightarrow \frac{d y}{d x}=-\frac{1}{\sin (y)}=-\sqrt{1-\cos ^{2}\left(\sqrt{1-x^{2}}\right.} \).

          Prove the formula for \( (d / d x)\left(\cos ^{-1}(x)\right) \) by the same method as for \( (d / d x)\left(\sin ^{-1}(x)\right) \).
Let \( y=\cos ^{-1}(x) \). Then \( \cos (y)= \) \( \square \) and \( 0 \leq y \leq \pi \Rightarrow-\sin (y) \frac{d y}{d x}=1 \Rightarrow \frac{d y}{d x}=-\frac{1}{\sin (y)}=-\sqrt{1-\cos ^{2}\left(\sqrt{1-x^{2}}\right.} \).

Prove the formula for (d / d x)(cos ^-1(x)) by the same method as for (d / d x)(sin ^-1(x)).
Let y=cos ^-1(x). Then cos (y)= □ and 0 ≤ y ≤π⇒-sin (y) (d y)/(d x)=1 ⇒(d y)/(d x)=-(1)/(sin (y))=-√(1-cos ^2(√(1-x^2).).

Added by Soledad M.

Calculus Early Transcendentals

James Stewart 7th Edition

Chapter 3

Instant Answer

Solved by Expert Jonathon Brumley

Step 1

This means that \( y \) is the angle whose cosine is \( x \). Therefore, by definition, \( \cos(y) = x \). Show more…

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Prove the formula for \( (d / d x)\left(\cos ^{-1}(x)\right) \) by the same method as for \( (d / d x)\left(\sin ^{-1}(x)\right) \). Let \( y=\cos ^{-1}(x) \). Then \( \cos (y)= \) \( \square \) and \( 0 \leq y \leq \pi \Rightarrow-\sin (y) \frac{d y}{d x}=1 \Rightarrow \frac{d y}{d x}=-\frac{1}{\sin (y)}=-\sqrt{1-\cos ^{2}\left(\sqrt{1-x^{2}}\right.} \).