- Each of the problems below involves a combination of two of the following: product rule, quotient rule, chain rule. Give them a shot!
Answer key note: the answers below are simplified, in some cases more so than I’d expect you to on a quiz or exam, but it is still good practice to try to simplify and see if you got the same thing I did.ans
- [latex]\frac{d}{dx} \ x^2 \ln(x) e^x[/latex]
[latex]x^2 \ln(x) e^x + 2x \ln(x) e^x + x e^x[/latex]ans
- [latex]\frac{d}{dx} \ (2x + 1)^5 (3x - 1)[/latex]
[latex]3 (2x + 1)^5 + 10(3x-1)(2x + 1)^4[/latex]ans
- [latex]\frac{d}{dx} \ \sqrt{\sin(x^2)}[/latex]
[latex]\frac{1}{2}(\sin(x^2))^{-1/2} \cos(x^2) 2x[/latex]ans
- [latex]\frac{d}{dx} \ e^{1/(x^2 - 1)}[/latex]
[latex]e^{\frac{1}{x^2 - 1}} \cdot \frac{-2x}{(x^2 - 1)^2}[/latex]ans
- [latex]\frac{d}{dx} \ \frac{x \ln(x)}{x + 1}[/latex]
[latex]\frac{\ln(x) + x + 1}{(x+1)^2}[/latex]ans
- [latex]\frac{d}{dx} \ \cfrac{\cfrac{1}{\ln(x)} + \ln(x)}{\ln(x)}[/latex]
[latex]\frac{-2}{x (\ln(x))^3}[/latex]ans
- [latex]\frac{d}{dx} \ e^{\cos(x) \sin(x)}[/latex]
[latex]e^{\cos(x) \sin(x)} \cdot ((\cos(x))^2 - (\sin(x))^2)[/latex]ans
- [latex]\frac{d}{dx} \ \sin\left(\frac{x \ln x}{e^x} \right)[/latex]
[latex]\cos\left( \frac{x \ln x}{e^x} \right) \cdot \frac{\ln(x) + 1 - x \ln(x)}{e^x}[/latex]ans
- [latex]\frac{d}{dx} \ x^2 \ln(x) e^x[/latex]