Calculus ADerivatives Of Logarithmic And Exponential Functions Problem 1

Find the derivative:

$$f(x)=x \ln (x)-x \quad f(x)=\sin (\ln (x))$$

$$f(x)=\sin (\ln (x)) \quad f(x)=\log _{10}\left(x^{3}+1\right)$$

$$f(x)=\sin x \ln (5 x) \quad y=\ln \left|+x-x^{3}\right|$$

$$g(x)=\ln \left(x+\sqrt{x^{2}-1}\right)$$

$$f(x)=x \ln (x)-x \longrightarrow f^{\prime}(x)=(1) \ln (x)+\frac{1}{x}(x)-1$$

$$f^{\prime}(x)=\ln (x)+1-1=\ln (x)$$

$$f(x)={\sin (\ln (x))}{1}=\cos (\ln (x)) \cdot \frac{1}{x}=\frac{\cos (\ln (x)}{x}$$

$$f(x)=\log _{10}\left(x^{3}+1\right)=\frac{3 x^{2}}{\left(x^{3}+1\right) \ln (10)}$$

$$\log _{a}(x)=\frac{1}{x \cdot \ln ({a})}$$

$$f(x)=\sin x \cdot \ln (5 x) \longrightarrow$$

$$f^{\prime}(x)=\cos x(\ln (5 x))+\frac{5}{5 x}(\sin x)$$

$$f^{\prime}(x)=\cos x \ln (5 x)+\frac{\sin x}{x}$$

$$y=\ln \left|+x-x^{3}\right| \Rightarrow y^{\prime}=\frac{1-3 x^{2}}{x-x^{3}}$$

$$g(x)=\ln \left(x+\sqrt{x^{2}-1}\right) \rightarrow g^{\prime}(x)=\frac{1+\frac{2 x}{2 \sqrt{x^{2}-1}}}{x+\sqrt{x^{2}-1}}$$

$$=\frac{1+\frac{x}{\sqrt{x^{2}-1}}}{x+\sqrt{x^{2} -1 }}$$

Differentiate $$f$$ and find the domain of $$f :$$

$$f(x)=\frac{x}{1-\ln (x-1)}$$

$$f^{\prime}(x)=\frac{(1)(1-\ln (x-1))-\left(-\frac{1}{x-1}\right)(x)}{(1-\ln (x-1))^{2}}$$

$$f^{\prime}(x)=\frac{1-\ln (x-1)+\frac{x}{x-1}}{(1-\ln (x-1))^{2}}$$

$$D f : 1-\ln (x-1)=0 \rightarrow \ln (x-1)=1$$

$$e^{\ln(x-1)}={e}^{1} \longrightarrow x-1=e$$

$$x=e+1 \quad \rightarrow x-1>0 \rightarrow x>1$$

$$D f=(1, \infty) \backslash\{e+1\}$$