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) \: \: (f(x + h) - f(x))/(h)`

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

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

`\qquad \sf  \dashrightarrow \: f {}^( \prime) (x) = \displaystyle \sf \lim_(h \to0) \: \: \frac{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`