Derivative practice

At this point, we’ve learned quite a few differentiation rules - two basic rules and all five rules of combination:

• Basic rules
  • The power rule: \(\displaystyle \frac{d}{dx} x^p = px^{p-1}\)
  • The natural exponential rule: \(\displaystyle \frac{d}{dx} e^x = e^x\)
• Rules of combination
  • The constant multiple rule: \(\displaystyle \frac{d}{dx} c f(x) = cf'(x)\)
  • The sum rule: \(\displaystyle \frac{d}{dx} (f(x)+g(x)) = f'(x)+g'(x)\)
  • The product rule: \(\displaystyle \frac{d}{dx} f(x)g(x) = f'(x)g(x) + f(x)g'(x)\)
  • The quotient rule: \[\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\]
  • The chain rule: \(\displaystyle \frac{d}{dx} f(g(x)) = f'(g(x))g'(x)\)

Developing the ability to apply these rules efficiently is a major objective for you in this course. So, let’s practice!

1. \(\displaystyle f(x)=(x+1)(3x^2+1)\)
2. \(\displaystyle f(x) = \frac{x+1}{3x^2 + 1}\)
3. \(\displaystyle f(x) = e^{3x}\)
4. \(\displaystyle f(x) = e^{3x}(x+1)^4\)
5. \(\displaystyle f(x) = e^x(x^2+1)(x^4+1)^4\)
6. \(\displaystyle f(x) = \sqrt{x^4+2}\)
7. \(\displaystyle f(x) = x^2\sqrt{x^4+2}\)