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The functions $f(x)$ and $g(x)$ are differentiable. The function $h(x)$ is defined as: $h(x)= \frac{f(x)}{g(x)}$ If $f(-1)= 6$ , $f'(-1)= -3$ , $g(-1)= 2$ , and $g'(-1)= 5$ , what is $h'(-1)$ ?

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Find derivatives of sine and cosine functions

Full solution

Q. The functions

f(x)

and

g(x)

are differentiable. The function

h(x)

is defined as:

h(x)= \frac{f(x)}{g(x)}

If

f(-1)= 6

,

f'(-1)= -3

,

g(-1)= 2

, and

g'(-1)= 5

, what is

h'(-1)

?

Use Quotient Rule: Step $1$ : Use the quotient rule to find $h'(x)$ . The quotient rule states that if $h(x) = \frac{f(x)}{g(x)}$ , then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ .
Substitute Given Values: Step $2$ : Substitute the given values into the derivative formula. Plugging in $f(-1) = 6$ , $f'(-1) = -3$ , $g(-1) = 2$ , and $g'(-1) = 5$ into the formula from Step $1$ : $\newline$ $h'(-1) = ((-3)(2) - (6)(5)) / (2)^2$ .
Perform Calculations: Step $3$ : Perform the calculations: $\newline$ $h'(-1) = (-6 - 30) / 4 = -36 / 4 = -9$ .

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