  ### Home > APCALC > Chapter 3 > Lesson 3.4.3 > Problem3-176

3-176.

Evaluate the following limits. (Hint: Review your solution for problem 3-92 first!)

These are both Hana's definition of the derivative. Instead of algebraically finding these limits (as you did in 3-175), you could deconstruct her definition and apply the power rule.

1. $\lim\limits_ { h \rightarrow 0 } \large\frac { ( x + h ) ^ { 5 } - x ^ { 5 } } { h }$

Hana's definition of the derivative:

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

What if $f(x)$? $f(x) = x^5$
What if $f ^\prime (x)$? $f ^\prime (x) = 5x^4$

$=\lim\limits_{h\rightarrow 0}\frac{(x+h)^5-x^5}{h}$

1. $\lim\limits_ { h \rightarrow 0 } \large\frac { 2 \sqrt { x + h } - 2 \sqrt { x } } { h }$

Hana's definition of the derivative:

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

What if $f(x)$? $f(x) =$ ___________
What if $f ^\prime (x)$? $f ^\prime (x) =$ ___________

$=\lim\limits_{h\rightarrow 0}\frac{2\sqrt{x+h}-2\sqrt{x}}{h}$