Differentiating Algebraic and Trigonometric Functions

Each of the following expressions can be thought of as the formula for a single-variable real-valued function. For each of these functions, write down an explicit formula for its derivative. To check your result you can use software capable of symbolic differentiation, e.g. WolframAlpha.

\(\displaystyle x^7+\tfrac{1}{x^7}+\sqrt[7]{x}+x^{0.7} \)
\(\displaystyle 17x^{\pi}\)
\(\displaystyle 1.7x^{\pi}\)
\(\displaystyle \frac{1}{\pi}\)
\(\displaystyle 1.7^{-1}x^{7.1}\)
\(\displaystyle 1.7^{2}x^{-7.1}\)
\(\displaystyle x^{2}x^{-7.1}\)
\(\displaystyle x^{2}17^{-7.1}\)
\(\displaystyle x^3+x^2+x+1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}\)
\(\displaystyle \bigl(x-17\bigr)^2 \)
\(\displaystyle \bigl(x-17^2\bigr) \)
\(\displaystyle \bigl(x-2\bigr)^{17} \)
\(\displaystyle \bigl(17x-2\bigr)^{17} \)
\(\displaystyle \bigl(17x^2-2\bigr)^{17} \)
\(\displaystyle \bigl(17x^2-2\bigr)\bigl(x^{17}-17x+2\bigr) \)
\(\displaystyle \tfrac{x}{17}\sqrt{x^2+x+1} \)
\(\displaystyle \sqrt{x^{17}+1}\sqrt{x-17} \)
\(\displaystyle \frac{17x^2-2}{x^{17}-17x+2} \)
\(\displaystyle \frac{1}{1+\frac{1}{x}} \)
\(\displaystyle \sqrt[7]{\sqrt[3]{\sqrt{x}}} \)
\(\displaystyle 71\sin(x) + 17\cos(x)\)
\(\displaystyle 17\sin(x)\cos(x)\)
\(\displaystyle \frac{\sin(x)}{x} \)
\(\displaystyle \frac{x}{\sin(x)} \)
\(\displaystyle \sin\bigl(\tfrac{1}{x}\bigr) \)
\(\displaystyle \sin\bigl(\cos(x)\bigr)\)
\(\displaystyle 17\sin^{17}(x)\)
\(\displaystyle \cos\bigl(x^{17}\bigr) \)
\(\displaystyle \cos^{17}(x) \)
\(\displaystyle 17\cos\bigl(17x\bigr) \)
\(\displaystyle \sec\bigl(x^{17}-x\bigr)\)
\(\displaystyle \csc\bigl(\sqrt{17x-1}\bigr)\)
\(\displaystyle \sqrt{x}\tan(x) \)
\(\displaystyle \sqrt{\tan(x)} \)
\(\displaystyle \cot\bigl(\sqrt{x}\bigr) \)
\(\displaystyle \tan\bigl(\sin(x)\bigr)\)
\(\displaystyle \tan^{17}\bigl(\sec(x)\bigr)\)
\(\displaystyle \frac{\sec(x)}{\tan(x)}-\sin(x) \)
\(\displaystyle \frac{\tan(x)}{\sin(x)} - \csc(x) \)
\(\displaystyle \frac{\cos^2(x)}{\sin(x)} \)
\(\displaystyle \tan(2x)\csc(17x)+1\)
\(\displaystyle \frac{\tan(2x)}{1+\csc(17x)}\)
\(\displaystyle \frac{1}{1+\frac{1}{\tan(17x)}}\)
\(\displaystyle \tan\bigl(\tan(x)\bigr)\)
\(\displaystyle \sec\bigl(\cot(x)\bigr) \)
\(\displaystyle \csc\bigl(\csc(x)\bigr)\)
\(\displaystyle \sec\bigl(\sin(x)\tan(x)\bigr)\)
\(\displaystyle \sin\Bigl(\cos\bigl(\tan(x)\bigr)\Bigr) \)
\(\displaystyle x^{17}\cot^{17}(x) \)
\(\displaystyle \frac{\sqrt{x^{17}}}{\cot^{17}(x)} \)
\(\displaystyle \biggl(\frac{\sqrt{x}}{\sec(x)}\biggr)^{17} \)
\(\displaystyle \frac{1}{x +2\csc(x)} \)
\(\displaystyle \frac{x +2\csc(x)}{x} \)
\(\displaystyle \frac{x +2\csc(x)}{x -2\csc(x)} \)
\(\displaystyle \frac{\sqrt[3]{x}}{\tan(x)} \)
\(\displaystyle \frac{\tan(x)}{\sqrt[3]{x}} \)
\(\displaystyle \sqrt[3]{\frac{\sec(x)}{x}} \)
\(\displaystyle \tan\Bigl(\cos\bigl(\cos(x)\bigr)\Bigr) \)
\(\displaystyle \sin\Bigl(\sec\bigl(\sec(x)\bigr)\Bigr) \)
\(\displaystyle \sec\Bigl(\tan\bigl(x^3\bigr)\Bigr) \)
\(\displaystyle \sin\Bigl(\sqrt{\cos(x)}\Bigr) \)
\(\displaystyle \sin\bigl(\sec^{17}(x)\tan(x)\bigr) \)
\(\displaystyle \cos\biggl(\frac{\cos(x)}{\cos\bigl(\cos(x)\bigr)}\biggr) \)
\(\displaystyle \sin\biggl(\frac{\sin^2(x)+\cos^2(x)}{\tan^2(x)+1}\biggr) \)

Each of the following equations involves an independent variable \(x\) and a variable \(y\) dependent on \(x.\) As exercise, for each of these equations, take the derivative with respect to \(x\) implicitly. Then if possible, express the derivative of \(y\) explicitly as a function of \(x\). You can use software capable of symbolic differentiation to check your formulas.

\(\displaystyle xy = 1 \)
\(\displaystyle (xy)^2 = 1 \)
\(\displaystyle (xy)^{17} = xy+1 \)
\(\displaystyle (x+y)^{17} = x+y+1 \)
\(\displaystyle (17x+71y)^2 = 1 \)
\(\displaystyle \bigl(17xy+71y^2\bigr)^2 = 1 \)
\(\displaystyle x^3+y^2+x = \frac{1}{y}+\frac{1}{x^2}+\frac{1}{y^3}\)
\(\displaystyle \frac{x}{y} = x+y \)
\(\displaystyle \frac{x^2}{y^3} = x^7+y \)
\(\displaystyle \frac{xy}{y+17} = x+y \)
\(\displaystyle \frac{1}{1+\frac{x}{y}} = 1\)
\(\displaystyle \sqrt{xy+2y} = 1 \)
\(\displaystyle 71\sin(x) = 17\cos(y)\)
\(\displaystyle \sin(x)\cos(y) = 1\)
\(\displaystyle \sin\bigl(x\cos(y)\bigr) = 1\)
\(\displaystyle 17\sin^{17}(y) = x\)
\(\displaystyle \sec\bigl(y^{17}-x\bigr) = 1\)
\(\displaystyle \sqrt{\cot(y)} = x\)
\(\displaystyle \tan\bigl(\sin(xy)\bigr)\)
\(\displaystyle \frac{1}{y+\frac{1}{\tan(17y)}}=x\)
\(\displaystyle \sin\Bigl(\cos\bigl(\sec(y)\bigr)\Bigr) = xy \)
\(\displaystyle \tan\Bigl(\cos\bigl(\cos(xy)\bigr)\Bigr) = y \)

Now suppose each of the previous equations involved an independent variable \(t\) (not explicitly present) and variables \(x\) and \(y\) each dependent on \(t.\) As exercise, for each of those equations equations, take the derivative with respect to \(t\) implicitly. You can use software capable of symbolic differentiation to check your formulas.