Solve the second order ordinary differential equation \[ \begin{array}{l}\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+l(l+1) y=0 \\ \text { and with the knowledge that Legendre's polynomials are defined such that } \mathrm{P}_{1}(\mathrm{x}=1)=1 \text {, show that } \\ \qquad P_{2}(x)=\frac{1}{2}\left(3 x^{2}-1\right)\end{array} \]

Evaluate the function. If the value is not a rational number, rouna (a) \( \operatorname{csch}^{-1}(3) \) (b) \( \operatorname{coth}^{-1}(6) \)

Consider the following. \[ g(x)=4 x \operatorname{sech}(x) \] Find \( g^{\prime}(x) \). \( g^{\prime}(x)=\square \) Find any critical numbers of \( g \). (Enter your answers as a comma-separated list. Round your answers to four decimal places.) Find the relative extrema of the function. (Round your answers to four decimal places.) relative maxima \( \quad(x, y)=(\square) \) relative minima \[ (x, y)=(\square) \]

Find the derivative of the function. \[ y=\tanh \left(6 x^{2}-7\right) \] \( y^{\prime}=\square \)

Find an equation of the tangent line to the graph of the function at the point \( (1,1) \). \[ y=x^{\cosh (x)} \] \( y=\square \)

Find the average rate of change of \( f(x)=-2 x^{2}-x \) from \( x=2 \) to \( x=6 \), Simplify your answer as much as possible.