Derivatives Practice Demo

Formulas and Supplemental Exercises

Formulas and rules...

\[ \mbox{Linearity:} \quad \dfrac{d}{dx}\Bigg[ f(x)+g(x) \Bigg] = f'(x)+g'(x) \quad \mbox{and} \quad \dfrac{d}{dx}\Bigg[ c\,f(x) \Bigg] = c\,f'(x) \qquad \qquad \dfrac{d}{dx}\Bigg[ mx+b \Bigg] =m \qquad \dfrac{d}{dx}\Bigg[ c \Bigg] =0 \] \[ \mbox{Power rule:} \quad \dfrac{d}{dx}\Bigg[ x^n \Bigg] = nx^{n-1} \qquad \qquad \dfrac{d}{dx}\Bigg[ b^x \Bigg] = b^x \,\ln(b) \qquad \dfrac{d}{dx}\Bigg[ e^x \Bigg] = e^x \qquad \dfrac{d}{dx}\Bigg[ \ln|x| \Bigg] = \dfrac{1}{x} \] \[ \mbox{Product rule:} \quad \dfrac{d}{dx}\Bigg[ f(x)g(x) \Bigg] = f'(x)g(x)+f(x)g'(x) \qquad \qquad \mbox{Quotient rule:} \quad \dfrac{d}{dx}\Bigg[ \dfrac{f(x)}{g(x)} \Bigg] = \dfrac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2} \] \[ \mbox{Chain rule:} \quad \dfrac{d}{dx}\Bigg[ f(g(x)) \Bigg] = f'(g(x))\,g'(x) \qquad \qquad \dfrac{d}{dx}\Bigg[ \sin(x) \Bigg] = \cos(x) \qquad \dfrac{d}{dx}\Bigg[ \cos(x) \Bigg] = -\sin(x) \] \[ \dfrac{d}{dx}\Bigg[ \tan(x) \Bigg] = \sec^2(x) \qquad \dfrac{d}{dx}\Bigg[ \sec(x) \Bigg] = \sec(x)\tan(x) \qquad \dfrac{d}{dx}\Bigg[ \cot(x) \Bigg] = -\csc^2(x) \qquad \dfrac{d}{dx}\Bigg[ \csc(x) \Bigg] = -\csc(x)\cot(x) \] \[ \dfrac{d}{dx}\Bigg[ \arcsin(x) \Bigg] = \dfrac{1}{\sqrt{1-x^2}} \qquad \dfrac{d}{dx}\Bigg[ \arccos(x) \Bigg] = -\dfrac{1}{\sqrt{1-x^2}} \qquad \dfrac{d}{dx}\Bigg[ \arctan(x) \Bigg] = \dfrac{1}{1+x^2} \] \[ \dfrac{d}{dx}\Bigg[ \sinh(x) \Bigg] = \cosh(x) \qquad \dfrac{d}{dx}\Bigg[ \cosh(x) \Bigg] = \sinh(x) \]

For completeness (but you don't need to know these)...

\[ \dfrac{d}{dx}\Bigg[ \mathrm{arcsec}(x) \Bigg] = \dfrac{1}{|x|\sqrt{x^2-1}} \qquad \dfrac{d}{dx}\Bigg[ \mathrm{arccsc}(x) \Bigg] = -\dfrac{1}{|x|\sqrt{x^2-1}} \qquad \dfrac{d}{dx}\Bigg[ \mathrm{arccot}(x) \Bigg] = -\dfrac{1}{1+x^2} \] \[ \dfrac{d}{dx}\Bigg[ \mathrm{tanh}(x) \Bigg] = \mathrm{sech}^2(x) \quad \dfrac{d}{dx}\Bigg[ \mathrm{sech}(x) \Bigg] = -\mathrm{sech}(x)\mathrm{tanh}(x) \quad \dfrac{d}{dx}\Bigg[ \mathrm{coth}(x) \Bigg] = -\mathrm{csch}^2(x) \quad \dfrac{d}{dx}\Bigg[ \mathrm{csch}(x) \Bigg] = -\mathrm{csch}(x)\mathrm{coth}(x) \] \[ \dfrac{d}{dx}\Bigg[ \mathrm{arcsinh}(x) \Bigg] = \dfrac{1}{\sqrt{x^2+1}} \qquad \dfrac{d}{dx}\Bigg[ \mathrm{arccosh}(x) \Bigg] = \dfrac{1}{\sqrt{x^2-1}} \qquad \dfrac{d}{dx}\Bigg[ \mathrm{arctanh}(x) \Bigg] = \dfrac{1}{1-x^2} \] \[ \dfrac{d}{dx}\Bigg[ \mathrm{arcsech}(x) \Bigg] = -\dfrac{1}{x\sqrt{1-x^2}} \qquad \dfrac{d}{dx}\Bigg[ \mathrm{arccsch}(x) \Bigg] = -\dfrac{1}{|x|\sqrt{x^2+1}} \qquad \dfrac{d}{dx}\Bigg[ \mathrm{arccoth}(x) \Bigg] = \dfrac{1}{1-x^2} \]

A Bit about Technology...

The back of this sheet contains a list of practice problems. It also contains links to solutions generated by Wolfram Alpha and SymboLab.

There are several major computer algebra systems (CAS for short). Many of our undergraduate math classes use a system called Maple. Another major CAS is Mathematica. Mathematica is part of what powers the online computational engine known as Wolfram Alpha.

We have used several demos powered by the SageMath CAS. While SAGE's user interface is not as polished as those of Maple and Mathematica, it is open source and free to use. Even Wolfram Alpha, while available freely, has locked featured behind its paywall.

Of course, Maple, Mathematica, and SAGE are not the only tools available. We have also seen Desmos which creates beautiful graphs (but does not do symbolic calculations). SymboLab provides yet another tool for doing mathematical computations.

SymboLab is much more limited than the CAS mentioned above, but it has an easy user interface and provides step-by-step solutions of basic problems in algebra and calculus including step-by-step derivative calculations. If you're still having trouble with derivative rules, I think you'll find SymboLab to be very helpful.

It is also worth mentioning that SAGE, Desmos, Wolfram Alpha, and SymboLab are all available for free via a web browser. Some of these have accompanying smartphone apps (which generally are not free). For example, Desmos is free while Alpha's is a few dollars. SymboLab's app is free but has locked features which require a payment to unlock. Personally, I'd just access these tools via a browser.

Practice Problems...

Compute the derivative. These problems begin with ``basics'' and then start adding rules and new functions. The problems toward the end are generally trickier than those at the beginning.

Try some basic simplification as well. Being able to simplify answers is important in many contexts -- that said -- simplification is not the focus of this exercise so don't waste a lot of time fighting with algebra.

Note: Clicking on "ALPHA" will take you Wolfram Alpha's solution, and clicking on "SYMBO" will take you to SymboLab's solution.

  1. ALPHA     SYMBO $\qquad \qquad y=\sqrt[3]{x}+\dfrac{1}{x}-\ln(x)+7x \qquad \qquad$

  2. ALPHA     SYMBO $\qquad \qquad y=\dfrac{3x^2-\sqrt{x}}{x^5}+\pi^{10} \qquad \qquad$

  3. ALPHA     SYMBO $\qquad \qquad y=3^x(x^5+9x-1) \qquad \qquad$

  4. ALPHA     SYMBO $\qquad \qquad y=\dfrac{x^2+3x}{3x^2-2x+1} \qquad \qquad$

  5. ALPHA     SYMBO $\qquad \qquad y=\dfrac{e^x+2x-1}{5x^3+x^2-12} \qquad \qquad$

  6. ALPHA     SYMBO $\qquad \qquad y=e^{2x}(x^3+6x-1)\ln(x) \qquad \qquad$

  7. ALPHA     SYMBO $\qquad \qquad y=x^5\sin(3x) \qquad \qquad$

  8. ALPHA     SYMBO $\qquad \qquad y=\dfrac{1}{\sqrt{x^6-5x^4+x^3-2}} \qquad \qquad$

  9. ALPHA     SYMBO $\qquad \qquad y=(x^2+\cos(x))^5 \qquad \qquad$

  10. ALPHA     SYMBO $\qquad \qquad y=99^{\mathrm{arctan}(x)} \qquad \qquad$

  11. ALPHA     SYMBO $\qquad \qquad y=x\sinh(x^2) \qquad \qquad$

  12. ALPHA     SYMBO $\qquad \qquad y=e^{x^2\sin(8x)} \qquad \qquad$

  13. ALPHA     SYMBO $\qquad \qquad y=\sinh^3(x)\sin^2(x) \qquad \qquad$

  14. ALPHA     SYMBO $\qquad \qquad y=\tan\left(\dfrac{x}{x+1}\right) \qquad \qquad$

  15. ALPHA     SYMBO $\qquad \qquad y= \dfrac{1+\csc(x^2)}{1-\cot(x^2)} \qquad \qquad$

  16. ALPHA     SYMBO $\qquad \qquad y=\tan^5(x^3) \qquad \qquad$

  17. ALPHA     SYMBO $\qquad \qquad y=\dfrac{\sec(9x)+1}{x^3e^{-x}} \qquad \qquad$

  18. ALPHA     SYMBO $\qquad \qquad y=\ln(\arcsin(x)+1) \qquad \qquad$

  19. ALPHA     SYMBO $\qquad \qquad y=\dfrac{\sin(x^3)+2x-1}{\mathrm{arctan}(x)} \qquad \qquad$

  20. ALPHA     SYMBO $\qquad \qquad y=\sqrt{x^{6}+e^{4x}+\sinh(5x+1)} \qquad \qquad$

  21. ALPHA     SYMBO $\qquad \qquad y=\cosh\left(\dfrac{x^2+1}{e^{7x}}\right) \qquad \qquad$

  22. ALPHA     SYMBO $\qquad \qquad y=\sec(\ln(\mathrm{arcsin}(5^x))) \qquad \qquad$

  23. ALPHA     SYMBO $\qquad \qquad y=\tan\left(\dfrac{5}{x}\right) \qquad \qquad$

  24. ALPHA     SYMBO $\qquad \qquad y=\ln\left(\dfrac{\sqrt[3]{7x^5+2x+1}}{x^{10}e^x}\right) \qquad \qquad$

  25. ALPHA     SYMBO $\qquad \qquad y=\ln\left(\dfrac{x^9(x^2+1)}{e^{3x+2}\sqrt{x}}\right) \qquad \qquad$

  26. ALPHA     SYMBO $\qquad \qquad y=\csc(5x)\,\mathrm{arccos}(x^3) \qquad \qquad$

  27. ALPHA     SYMBO $\qquad \qquad y=\ln(x^7\sin(3x)) \qquad \qquad$

  28. ALPHA     SYMBO $\qquad \qquad y=\cos(\cot(\sqrt{1+x^2})) \qquad \qquad$

  29. ALPHA     SYMBO $\qquad \qquad y=\sqrt[3]{x^2\ln(x^{55})} \qquad \qquad$

  30. ALPHA     SYMBO $\qquad \qquad y=(\csc(2x)+\sec(3x))^5 \qquad \qquad$



Given a function (specified in the box labeled "$f(x)=$"), this demo computes as many derivatives as requested (in the other box). The dropdown list contains the list of practice problems from above. If you change a box entry, simply click outside the box to have sage recompute the plots and displayed values.

The computation above is powered by SageMath. The Sage code is embedded in this webpage's html file. To view the code instruct your browser to show you this page's source. For example, in Chrome, right-click and choose "View page source".