91Ó°ÊÓ

Differentiate. $$ \cos (\sec (\sin 2 x)) $$

Differentiate. $$ \tan (\cot (\sec 3 x)) $$

Differentiate. $$ \sqrt[7]{\csc ^{3}\left(\frac{5 \pi}{6}+2.389\right)} $$

The Sum Identity for the sine function is $$\sin (a+x)=\sin a \cos x+\cos a \sin x$$ Differentiate both sides, thinking of \(a\) as a constant and \(x\) as the variable. Simplify your answer and show that you get the Sum Identity for the cosine function.

The sum formula [or the cosine function is $$\cos (a+x)=\cos a \cos x-\sin a \sin x$$ Differentiate both sides, thinking of \(a\) as a constant and \(x\) as the variable. Simplify your answer and show that you get the sum formula for the sine function.

Let \(f(x)=\frac{x^{2}}{x^{2}-1}\) and \(g(x)=\frac{1}{x^{2}-1}\). a) Compute \(f^{\prime}(x)\). b) Compute \(g^{\prime}(x)\). c) Compare your answers in parts (a) and (b) and explain.

Let \(f(x)=\sin ^{2} x+\cos ^{2} x\). a) Compute \(f^{\prime}(x)\). Don't simplify \(f(x)\) before differ entiating, but simplify your answer. b) Explain your answer.

Graph \(f\) and \(f^{\prime}\) over the given interval. Then estimate points at which the tangent line is horizontal. $$ f(x)=\sqrt{6 x^{3}-3 x^{2}-48 x+45} ;[-5,5] $$

Let \(f(x)=\tan ^{2} x\) and \(g(x)=\sec ^{2} x\) a) Compute \(f^{\prime}(x)\). b) Compute \(g^{\prime}(x)\). c) Compare your answers to parts (a) and (b) and explain.

Find the derivative of each of the following functions analytically. Then use a grapher to check the results. $$ f(x)=x \sqrt{4-x^{2}} $$