Home > APCALC > Chapter 5 > Lesson 5.3.1 > Problem5-112

5-112.

Differentiate each of the following functions.

1. $y=\large\frac{-2}{x^3}$

Do NOT use the quotient rule (though it will work). This is a Power Rule problem: $y = −2x^{−3}$. Reserve the Quotient Rule for situations in which there are variables in BOTH the numerator and the denominator.

2. $y = \operatorname{tan}(w^2 + 3)$

Chain Rule.

3. $y = -5(x - 3)^8$

This is not technically a Chain Rule problem, though you may use it.

4.  $y=\large\frac{\cos(x)}{(x-3)^8}$

This calls for the Quotient Rule.

$y'(x)=\frac{-\text{sin}(x)(x-3)-8\text{cos}(x)}{(x-3)^{9}}$

5. $y = x \operatorname{tan}(x^2 + 3)$

Product Rule & Chain Rule combined.

6. $y = \large\frac { 3 } { \operatorname { cos } t }$

Refer to the hint in part (a). And recall that the reciprocal of $\operatorname{cos} x$ is $\operatorname{sec} x$.