2.11 More Problems

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Use the differentiation rules to compute derivatives

Compute $\frac {d}{dx}x^3\ln (x)$

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

Compute $\frac {d}{dx} x \tan ^{-1} (x)$

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

Compute $\frac {d}{dx} \ln (\csc (x))$

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

Compute $\frac {d}{du} \sec (3u)$

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

Compute $\frac {d}{dx} e^{2x} \cos (3x)$

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

Compute $\frac {d}{dx} 2^{\cot (x)}$

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

Compute $\frac {d}{dx} \frac {e^{3x}}{x^2 + 1}$

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

Compute $\frac {d}{du} \tan ^{-1}(u^3)$

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

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

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

Compute $\frac {d}{dx} \sin (\tan (x))$

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

Compute $\frac {d}{dx} \sin (x\ln (x))$

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

Compute $\frac {d}{dx} \ln (x\cos (x))$

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

Compute $\frac {d}{dx} \tan ^3(5x)$

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

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

$f'(0) = \answer [given]{0}$

Compute $f'(0)$ if $f(x)=$ $\frac {d}{dx} \tan ^{-1}(e^x)$

$f'(0) = \answer [given]{1}$