So far we have emphasized derivatives of particular functions, but sometimes we want to look at the  derivatives of a whole family of functions.  In problems  49 –71, the letters A – D represent constants and the given formulas describe families of functions. 

For problems 49-65, calculate \mathrm{y}^{\prime}=\frac{\mathrm{d} \mathrm{y}}{\mathrm{d} \mathrm{x}}.


49. y=A x^{3}+B x^{2}+C


51. y=\sin \left(A x^{2}+B\right)


53. \mathrm{y}=\sqrt{\mathrm{A}+\mathrm{Bx}^{2}}


55. y=A-\cos (B x)


57. y=\cos \left(A x^{2}+B\right)


59. y=x \cdot e^{B x}


61. y=e^{A x}-e^{-A x}


63. y=\frac{A x}{\sin (B x)}


65. \mathrm{y}=\frac{\mathrm{Ax}+\mathrm{B}}{\mathrm{Cx}+\mathrm{D}}

In problems 67-71, (a) find y ', (b) find the value(s) of x so that y^{\prime}=0, and (c) find y " Typically your answer in part (b) will contain As, Bs and (sometimes) Cs.

67. \mathrm{y}=\mathrm{Ax}(\mathrm{B}-\mathrm{x})=\mathrm{ABx}-\mathrm{Ax}^{2}


69. \mathrm{y}=\mathrm{Ax}^{2}(\mathrm{~B}-\mathrm{x})=\mathrm{ABx}^{2}-\mathrm{Ax}^{3}


71. \mathrm{y}=\mathrm{Ax}^{3}+\mathrm{Bx}^{2}+\mathrm{C}

Use the given differentiation patterns to differentiate the composite functions in problems 73-83. We have not derived the derivatives for these functions (yet), but if you are handed the derivative pattern for a function then you should be able to take derivatives of a composition involving that function.

Given:

\begin{align*}\mathbf{D}(\arctan (x))=\frac{1}{1+x^{2}}, D(\arcsin (x))=\frac{1}{\sqrt{1-x^{2}}}, \quad \mathbf{D}(\ln (x))=\frac{1}{x}\end{align*}


73. \frac{\mathbf{d}}{\mathbf{d x}}\left(\arctan \left(x^{2}\right)\right)


75. \mathbf{D}\left(\arctan \left(\mathrm{e}^{\mathrm{x}}\right)\right)


77. \quad D\left(\arcsin \left(x^{3}\right)\right)


79. \frac{\mathbf{d}}{\mathbf{d t}}\left(\arcsin \left(\mathrm{e}^{\mathrm{t}}\right)\right)


81. \frac{\mathbf{d}}{\mathbf{d x}}(\ln (\sin (x)))


83. \frac{\mathbf{d}}{\mathrm{ds}}\left(\ln \left(\mathrm{e}^{\mathrm{s}}\right)\right)