Question

What is the slope of y = 6 ^ x when x=2?
The formula for the slope is________for h close to 0 (but not equal 0)

The best estimate for the slope is_____

Answer 1

${ \qquad\qquad\huge\underline{{\sf Answer}}}$

Here we go ~

$\qquad \sf  \dashrightarrow \: f(x)= {6}^{x}$

we need to find f'(2) = ??

$\qquad \sf  \dashrightarrow \: f {}^{ \prime} (x) = \displaystyle \sf \lim_{h \to0} \: \: \dfrac{f(x + h) - f(x)}{h}$

$\qquad \sf  \dashrightarrow \: f {}^{ \prime} (x) = \displaystyle \sf \lim_{h \to0} \: \: \dfrac{6 {}^{x + h} - 6 {}^{x} }{h}$

$\qquad \sf  \dashrightarrow \: f {}^{ \prime} (x) = \displaystyle \sf \lim_{h \to0} \: \: \dfrac{6 {}^{x + h} - 6 {}^{x} }{h}$

$\qquad \sf  \dashrightarrow \: f {}^{ \prime} (x) = \displaystyle \sf \lim_{h \to0} \: \: \dfrac{6 {}^{x }( 6 {}^{h} - 1)}{h}$

$\qquad \sf  \dashrightarrow \: f {}^{ \prime} (x) =\: 6 {}^{x} \: log_{e}(6)$

Now, plug in 2 for x ~

$\qquad \sf  \dashrightarrow \: f {}^{ \prime} (2) =\: 6 {}^{2} \sdot log_{e}(6)$

$\qquad \sf  \dashrightarrow \: f {}^{ \prime} (2) =\: 36 \sdot (1.79)$

$\qquad \sf  \dashrightarrow \: f {}^{ \prime} (2) =64.44$