3.3 HW Problems

Table of Contents:

Powers, Multiples, Sums, and Differences

Derivative of a Constant Function

• A basic rule of differentiation is that the derivative of every constant function is zero.

$$ \frac{df}{dx}=\frac{d}{dx}(c)=0 $$

Proof:

$$ f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim_{h \rightarrow 0} \frac{c-c}{h}= \lim_{h \rightarrow 0}0=0 $$

Derivative of a Positive Integer Power

• If n is a positive integer, then

$$ \frac{d}{dx}x^n=nx^{n-1} $$

Power Rule (general version) → $f(x)=n^x$

• If n is any real number, then

$$ \frac{d}{dx}x^n=nx^{n-1} $$

Example:

• Differentiate the following powers of x.
  • $x^3$
  • $x^{2/3}$
  • $x^{\sqrt{2}}$
  • $\frac{1}{x^4}$
  • $x^{-4/3}$
  • $\sqrt{x^{2+\pi}}$