Differentiating Algebraic and Trigonometric Functions

\(\displaystyle \bigl( k f\bigr)' = kf' \text{ for } k \in \mathbf{R}\)
\(\displaystyle \bigl( f \pm g \bigr)' = f' \pm g' \)
\(\displaystyle \bigl( f \circ g \bigr)' = \bigl(f'\circ g\bigr)(g') \)
\(\displaystyle \bigl( f g \bigr)' = f'g + fg' \)
\(\displaystyle \biggl( \frac{f}{g} \biggr)' = \frac{f'g - fg'}{g^2} \)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \bigl(x^n\bigr) = nx^{n-1} \text{ for } n\neq 0\)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \bigl(\sin(x)\bigr) = \cos(x) \)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \bigl(\sec(x)\bigr) = \sec(x)\tan(x) \)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \bigl(\tan(x)\bigr) = \sec^2(x) \)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \bigl(\cos(x)\bigr) = -\sin(x) \)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \bigl(\csc(x)\bigr) = -\csc(x)\cot(x) \)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \bigl(\cot(x)\bigr) = -\csc^2(x) \)

Explicit Differentiation

Each of the following expressions of a variable \(x\) can be thought of as the formula for a real-valued function that’s continuous on some open subset of the real numbers. As exercise, for each of these functions, write down an explicit formula for its derivative.

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

Implicit Differentiation

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 \(x\).

\(\displaystyle xy = 1 \)
\(\displaystyle (xy)^2 = 1 \)
\(\displaystyle (xy)^2 = xy \)
\(\displaystyle (x+y)^2 = x+y \)
\(\displaystyle (17x+71y)^2 = 1 \)
\(\displaystyle \bigl(17xy+71y^2\bigr)^2 = 1 \)
\(\displaystyle \frac{x}{y} = x+y \)
\(\displaystyle \frac{xy}{y+17} = x+y \)