91Ó°ÊÓ

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ y=x+\cot x $$

Think About It Describe the relationship between the rate of change of \(y\) and the rate of change of \(x\) in each expression. Assume all variables and derivatives are positive. \(\begin{array}{llll}{\text { (a) } \frac{d y}{d t}=3 \frac{d x}{d t}} & {\text { (b) } \frac{d y}{d t}=x(L-x) \frac{d x}{d t},} & {0 \leq x \leq L}\end{array}\)

In Exercises 39–52, find the derivative of the function. $$ f(x)=\frac{x^{3}-3 x^{2}+4}{x^{2}} $$

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ g(t)=\sqrt[4]{t}+6 \csc t $$

Sketching a Graph Sketch a graph of a function whose derivative is always negative. Explain how you found the answer.

Finding a Derivative In Exercises \(43-64\) , find the derivative of the function. $$ g(x)=5 \tan 3 x $$

Find \(d^{2} y / d x^{2}\) implicitly in terms of \(x\) and \(y\). \(x^{2}+y^{2}=4\)

In Exercises 39–52, find the derivative of the function. $$ h(x)=\frac{4 x^{3}+2 x+5}{x} $$

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ h(x)=\frac{1}{x}-12 \sec x $$

Find \(d^{2} y / d x^{2}\) implicitly in terms of \(x\) and \(y\). \(x^{2} y-4 x=5\)