Each of the following expressions can be thought of as a formula for the derivative of a function. For each expression, write down a potential formula for that function. If the expression doesn’t immediately lend itself to antidifferentiation, try to rewrite it in an more amenable form. Recall that a proposed antiderivative can be verified by differentiating it, checking that the result is identical to the original expression.
\(\displaystyle 7x^{6}\)
\(\displaystyle x^{6}\)
\(\displaystyle \tfrac{1}{7}x^{6}\)
\(\displaystyle x^4+x^3+x^2+x+1+\tfrac{1}{x^2}+\tfrac{1}{x^3}+\tfrac{1}{x^4} \)
\(\displaystyle \sqrt{x} + 2\sqrt[3]{x} + \sqrt{5}{x^6} \)
\(\displaystyle \frac{1}{\sqrt[3]{x}} \)
\(\displaystyle 7x^{\pi}\)
\(\displaystyle \sin(x)\)
\(\displaystyle \cos(x)\)
\(\displaystyle \tan(x)\sec(x)\)
\(\displaystyle \sec^2(x)\)
\(\displaystyle \sin(x)-\cos(x)\)
\(\displaystyle \sec(x)\bigl(\tan(x)+\sec(x)\bigr)\)
\(\displaystyle \bigl(7-x^3\bigr)^2\)
\(\displaystyle \bigl(x-5\bigr)\bigl(1-2x\bigr)\)
\(\displaystyle \sqrt[3]{x}\bigl(x^2-11\bigr)\)
\(\displaystyle \sqrt{x^2-8x+16}\)
\(\displaystyle \frac{x^3-x}{\sqrt{x}}\)
\(\displaystyle \frac{1+\frac{1}{x}}{x^2} \)