2.11 More Problems

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Use the differentiation rules to compute derivatives

(problem 1) Compute $\frac {d}{dx}\left [x^3\ln (x)\right ]$

The derivative of $x^3\ln (x)$ with respect to $x$ is $\answer {x^2 + 3x^2\ln (x)}$

(problem 2) Compute $\frac {d}{dx} \left [x \tan ^{-1} (x)\right ]$

The derivative of $x \tan ^{-1} (x)$ with respect to $x$ is $\answer {\tan ^{-1}(x) + \frac {x}{1+x^2}}$

(problem 3) Compute $\frac {d}{dx}\left [ \ln (\csc (x))\right ]$

The derivative of $\ln (\csc (x))$ with respect to $x$ is $\answer {-\cot (x)}$

Here is a video of problems 1, 2 and 3

(problem 4) Compute $\frac {d}{du} \left [\sec (3u)\right ]$

The derivative of $\sec (3u)$ with respect to $u$ is $\answer {3\sec (3u)\tan (3u)}$

(problem 5) Compute $\frac {d}{dx} \left [e^{2x} \cos (3x)\right ]$

The derivative of $e^{2x} \cos (3x)$ with respect to $x$ is $\answer {e^{2x}(2\cos (3x)-3\sin (3x))}$

(problem 6) Compute $\frac {d}{dx} \left [2^{\cot (x)}\right ]$

The derivative of $2^{\cot (x)}$ with respect to $x$ is $\answer {-2^{\cot (x)}\ln (2)\csc ^2(x)}$

Here is a video of problems 5 and 6

(problem 7) Compute $\frac {d}{dx} \left (\frac {e^{3x}}{x^2 + 1}\right )$

The derivative of $\frac {e^{3x}}{x^2 + 1}$ with respect to $x$ is $\answer {\frac {e^{3x}(3x^2 -2x +3)}{(x^2 + 1)^2}}$

(problem 8) Compute $\frac {d}{du} \left [\tan ^{-1}(u^3)\right ]$

The derivative of $\tan ^{-1}(u^3)$ with respect to $u$ is $\answer {\frac {3u^2}{1+u^6}}$

(problem 9) Compute $\frac {d}{dx} \left [\tan \left (\frac {2x-5}{3x+1}\right )\right ]$

The derivative of $\tan \left (\frac {2x-5}{3x+1}\right )$ with respect to $x$ is $\answer {\sec ^2(\frac {2x-5}{3x+1})\frac {17}{(3x+1)^2}}$

(problem 10) Compute $\frac {d}{dx} \left [\sin (\tan (x))\right ]$

The derivative of $\sin (\tan (x))$ with respect to $x$ is $\answer {\cos (\tan x)\sec ^2 x}$

(problem 11) Compute $\frac {d}{dx} \left [\sin (x\ln (x))\right ]$

The derivative of $\sin (x\ln (x))$ with respect to $x$ is $\answer {\cos (x\ln x)(1+\ln x)}$

(problem 12) Compute $\frac {d}{dx} \left [\ln (x\cos (x))\right ]$

The derivative of $\ln (x\cos (x))$ with respect to $x$ is $\answer {\frac {\cos x - x\sin x}{x\cos x}}$

Here are video solutions of problems 10, 11 and 12:

(problem 13) Compute $\frac {d}{dx} \left [\tan ^3(5x)\right ]$

The derivative of $\tan ^3(5x)$ with respect to $x$ is $\answer {15\tan ^2(5x)\sec ^2(5x)}$

(problem 14) Compute $f'(0)$ if $\displaystyle f(x) = \left (\frac {x^3 + 2x}{5x^2 -3}\right )^5$

$f'(0) = \answer {0}$

(problem 15) Compute $f'(0)$ if $\displaystyle f(x)=\tan ^{-1}(e^x)$

$f'(0) = \answer {1/2}$

2026-02-26 17:01:48