  ### Home > APCALC > Chapter 6 > Lesson 6.1.2 > Problem6-26

6-26.

Examine the following derivatives. Consider the multiple tools available for differentiation and use the best strategy for each part. Evaluate each derivative and briefly describe your method.

1. $\frac { d } { d x } [ \operatorname { sin } ( x ) \cdot e ^ { x } ]$

The Product Rule

If $f^\prime\left(x\right)$ and $g^\prime\left(x\right)$ exist and $j\left(x\right) = f\left(x\right) · g\left(x\right)$, then $j^\prime\left(x\right) = f ^\prime\left(x\right) · g\left(x\right) + f\left(x\right) · g^\prime\left(x\right)$.

2. $\frac { d } { d z } [ \frac { 6 z + 1 } { 3 z - 2 } ]$

The Quotient Rule

If $f ^\prime\left(x\right)$ and $g^\prime\left(x\right)$ exist and $h(x)=\frac{f(x)}{g(x)}$ where $g\left(x\right) ≠ 0$, then $h^\prime(x) =\frac { f ^ { \prime } ( x ) g ( x ) - f ( x ) g ^ { \prime } ( x ) } { ( g ( x ) ) ^ { 2 } }$ .

3. $\frac { d } { d t } [ \operatorname { tan } ( t ) \cdot \operatorname { cos } ( t ) ]$

$\text{tan}(t)=\frac{\text{sin}(t)}{\text{cos}(t)}$