Number of correct answers: 0
\(\displaystyle\dfrac{d}{dx}\Big(x^{10}\Big)\)
Ans: \(10x^{9}\)
\(\displaystyle\dfrac{d}{dx}\Big(\frac{1}{x^5}\Big)\)
Ans: \(\dfrac{-5}{x^6}\)
\(\displaystyle\dfrac{d}{dx}\Big(\sqrt{x}\Big)\)
Ans: \(\dfrac{1}{2\sqrt{x}}\)
\(\displaystyle\dfrac{d}{dx}\Big(\frac{1}{\sqrt{x}}\Big)\)
Ans: \(-\frac{1}{2}x^{-3/2}\)
\(\displaystyle\dfrac{d}{dx}\Big(\sin(2x)\Big)\)
Ans: \(2\cos(2x)\)
\(\displaystyle\dfrac{d}{dx}\Big(\cos(3x)\Big)\)
Ans: \(-3\sin(3x)\)
\(\displaystyle\dfrac{d}{dx}\Big(\tan(5x)\Big)\)
Ans: \(5\sec^2(5x)\)
\(\displaystyle\dfrac{d}{dx}\Big(\sec(3x)\Big)\)
Ans: \(3\sec(3x)\tan(3x)\)
\(\displaystyle\dfrac{d}{dx}\Big(\arcsin(2x)\Big)\)
Ans: \(\dfrac{2}{\sqrt{1-4x^2}}\)
\(\displaystyle\dfrac{d}{dx}\Big(\arctan(4x)\Big)\)
Ans: \(\dfrac{4}{1+16x^2}\)
\(\displaystyle\dfrac{d}{dx}\Big(e^{-3x}\Big)\)
Ans: \(-3e^{-3x}\)
\(\displaystyle\dfrac{d}{dx}\Big(\ln|x|\Big)\)
Ans: \(\dfrac{1}{x}\)
\(\displaystyle\dfrac{d}{dx}\Big(\ln(1+3x^2)\Big)\)
Ans: \(\dfrac{6x}{1+3x^2}\)
\(\displaystyle\dfrac{d}{dx}\Big(\ln|\sin(3x)|\Big)\)
Ans: \(3\csc(3x)\)
\(\displaystyle\dfrac{d}{dx}\Big(\ln|\sec(4x)|\Big)\)
Ans: \(4\tan(4x)\)
\(\displaystyle\dfrac{d}{dx}\Big(x\cdot\sin(3x)\Big)\)
Ans: \(\sin(3x) + 3x\cos(3x)\)
\(\displaystyle\dfrac{d}{dx}\Big(x^2\cdot\sec(3x)\Big)\)
Ans: \(2x\sec(3x) + 3x^2\sec(3x)\tan(3x)\)
\(\displaystyle\dfrac{d}{dx}\Big(x\cdot\arctan(2x)\Big)\)
Ans: \(\arctan(2x)+\dfrac{2x}{1+4x^2}\)
\(\displaystyle\dfrac{d}{dx}\Big(\arcsin(2\sqrt{x})\Big)\)
Ans: \(\dfrac{1}{\sqrt{1-4x}\,\sqrt{x}}\)
\(\displaystyle\dfrac{d}{dx}\Big(\arctan(3\sqrt{x})\Big)\)
Ans: \(\dfrac{3}{(1+9x)\,2\sqrt{x}}\)
\(\displaystyle\dfrac{d}{dx}\Big(\dfrac{\cos(x)}{x}\Big)\)
Ans: \(\dfrac{-\sin(x)\, x - \cos(x)}{x^2}\)
\(\displaystyle\dfrac{d}{dx}\Big(\dfrac{x}{\ln(x)}\Big)\)
Ans: \(\dfrac{\ln(x) - 1}{(\ln(x))^2}\)
\(\displaystyle\dfrac{d}{dx}\Big(\dfrac{\tan(2x)}{3x+1}\Big)\)
Ans: \(\dfrac{2\sec^2(2x)(3x+1) - 3\tan(2x)}{(3x+1)^2}\)
\(\displaystyle\dfrac{d}{dx}\Big(\arctan(e^{2x})\Big)\)
Ans: \(\dfrac{2e^{2x}}{1+e^{4x}}\)
\(\displaystyle\dfrac{d}{dx}\Big(\sin(x^2)\Big)\)
Ans: \(2x\,\cos(x^2)\)
\(\displaystyle\dfrac{d}{dx}\Big(2x\cos(x^2)\Big)\)
Ans: \(2\cos(x^2)-4x^2\sin(x^2)\)
\(\displaystyle\dfrac{d}{dx}\Big(\cos^2(x)\Big)\)
Ans: \(-2\cos(x)\sin(x)\)
\(\displaystyle\dfrac{d}{dx}\Big(\sec^3(2x)\Big)\)
Ans: \(6\sec^3(2x)\tan(2x)\)
\(\displaystyle\dfrac{d}{dx}\Big(\dfrac{e^{3x}}{\sin(2x)}\Big)\)
Ans: \(\dfrac{3e^{3x}\sin(2x) - 2e^{3x}\cos(2x)}{\sin^2(2x)}\)
\(\displaystyle\dfrac{d}{dx}\Big(\cos(\ln(x))\Big)\)
Ans: \(-\dfrac{\sin(\ln(x))}{x}\)
\(\displaystyle\dfrac{d}{dx}\Big(e^{-2x}\tan(3x)\Big)\)
Ans: \(-2e^{-2x}\tan(3x)+3e^{-2x}\sec^2(3x)\)
\(\displaystyle\dfrac{d}{dx}\Big(-xe^{-x^2}\Big)\)
Ans: \(-e^{-x^2}+2x^2e^{-x^2}\)
\(\displaystyle\dfrac{d}{dx}\Big(\frac{2x^2+3}{3x^2+2}\Big)\)
Ans: \(\dfrac{-10x}{(3x^2+2)^2}\)
\(\displaystyle\dfrac{d}{dx}\Big(\dfrac{2-\ln(x)}{1+3\ln(x)}\Big)\)
Ans: \(\dfrac{-7}{x\,(1+3\ln(x))^2}\)
\(\displaystyle\int x^{10}\ dx\)
Ans: \(\frac{1}{11}x^{11}+C\)
\(\displaystyle\int \frac{1}{x^{10}}\ dx\)
Ans: \(\dfrac{-1}{9x^{9}}+C\)
\(\displaystyle\int \sqrt{x}\ dx\)
Ans: \(\frac{2}{3}x^{3/2}+C\)
\(\displaystyle\int \frac{1}{\sqrt{x}}\ dx\)
Ans: \(2\sqrt{x}+C\)
\(\displaystyle\int \cos(3x)\ dx\)
Ans: \(\frac{1}{3}\sin(3x)+C\)
\(\displaystyle\int \cos(\frac{x}{3})\ dx\)
Ans: \(3\sin(\frac{x}{3})+C\)
\(\displaystyle\int \sin(2x)\ dx\)
Ans: \(-\frac{1}{2}\cos(2x)+C\)
\(\displaystyle\int \sec^2(3x)\ dx\)
Ans: \(\frac{1}{3}\tan(3x)+C\)
\(\displaystyle\int \sec^2(\frac{x}{4})\ dx\)
Ans: \(4\tan(\frac{x}{4})+C\)
\(\displaystyle\int \frac{1}{x}\ dx\)
Ans: \(\ln|x|+C\)
\(\displaystyle\int \frac{1}{\sqrt{1-x^2}}\ dx\)
Ans: \(\arcsin(x)+C\)
\(\displaystyle\int \frac{1}{1+x^2}\ dx\)
Ans: \(\arctan(x)+C\)
\(\displaystyle\int \frac{1}{1+4x^2}\ dx\)
Ans: \(\frac{1}{2}\arctan(2x)+C\)
\(\displaystyle\int \frac{1}{\sqrt{1-9x^2}}\ dx\)
Ans: \(\frac{1}{3}\arcsin(3x)+C\)
\(\displaystyle\int \frac{1}{x^2}\ dx\)
Ans: \(-\dfrac{1}{x}+C\)
\(\displaystyle\int e^{x}\ dx\)
Ans: \(e^{x}+C\)
\(\displaystyle\int e^{-x}\ dx\)
Ans: \(-e^{-x}+C\)
\(\displaystyle\int e^{3x}\ dx\)
Ans: \(\frac{1}{3}e^{3x}+C\)
\(\displaystyle\int e^{x/4}\ dx\)
Ans: \(4e^{x/4}+C\)
\(\displaystyle\int \frac{3x}{x^2}\ dx\)
Ans: \(3\ln|x|+C\)
\(\displaystyle\dfrac{d}{dx}\Big(\int_{-2}^{x^2}\sqrt{4+t^4}dt\Big)\)
Ans: \(2x\sqrt{4+x^8}\)
\(\displaystyle\dfrac{d}{dx}\Big(\int_{0}^{\sqrt{x}}e^{t^2}dt\Big)\)
Ans: \(\frac{e^{x}}{2\sqrt{x}}\)
\(\displaystyle\dfrac{d}{dx}\Big(\int_{1}^{3x}\sin(e^t)dt\Big)\)
Ans: \(3\sin(e^{3x})\)
\(\displaystyle\dfrac{d}{dx}\Big(\int_{-2}^{\cos(x)}\sqrt{4+t^4}dt\Big)\)
Ans: \(-\sin(x)\sqrt{4+\cos^4(x)}\)
\(\displaystyle\int \dfrac{1}{4+x^2}\ dx\)
Ans: \(\frac{1}{2}\arctan(x/2)+C\)
\(\displaystyle\int \dfrac{1}{\sqrt{9-x^2}}\ dx\)
Ans: \(\arcsin(x/3)+C\)