### Home > APCALC > Chapter 6 > Lesson 6.1.2 > Problem6-24

6-24.

First, decide if differentiating the following expressions requires the Product Rule, the Quotient Rule, the Chain Rule, or a combination of these rules. Then, evaluate the following derivatives.

1. $\frac { d } { d x }$$[ f ( t ^ { 2 } ) ]$

The Chain Rule

The Chain Rule allows us to differentiate composite functions.

If $h\left(x\right) = f\left(g\left(x\right)\right)$, then $h^\prime\left(x\right) = f ^\prime\left(g\left(x\right)\right) · g^\prime\left(x\right)$.

2. $\frac { d } { d x }$$[ x \cdot h ( 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 $j^\prime\left(x\right) = f ^\prime\left(x\right) · g\left(x\right) + f\left(x\right) · g^\prime\left(x\right)$.

3. $\frac { d } { d x }$$[ t \cdot h ( t ^ { 2 } ) ]$