# How do you find the derivative using the difference quotient #f(x) = 5 / (x^2 + 7)#?

The quotient rule is:

So:

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The details depend on which difference quotient you use :

or,

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To find the derivative using the difference quotient for ( f(x) = \frac{5}{x^2 + 7} ), first, apply the definition of the difference quotient. Then simplify the expression and take the limit as ( h ) approaches ( 0 ). This will give you the derivative of ( f(x) ) with respect to ( x ).

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To find the derivative of ( f(x) = \frac{5}{x^2 + 7} ) using the difference quotient, we start by using the definition of the derivative:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Substitute the function ( f(x) = \frac{5}{x^2 + 7} ) into the difference quotient:

[ f'(x) = \lim_{h \to 0} \frac{\frac{5}{(x + h)^2 + 7} - \frac{5}{x^2 + 7}}{h} ]

Now, let's simplify this expression:

[ f'(x) = \lim_{h \to 0} \frac{5(x^2 + 7) - 5((x + h)^2 + 7)}{h((x^2 + 7)((x + h)^2 + 7))} ]

[ f'(x) = \lim_{h \to 0} \frac{5x^2 + 35 - (5x^2 + 10hx + 5h^2 + 35)}{h((x^2 + 7)((x + h)^2 + 7))} ]

[ f'(x) = \lim_{h \to 0} \frac{-10hx - 5h^2}{h((x^2 + 7)((x + h)^2 + 7))} ]

Now, we can cancel out ( h ) from the numerator and denominator:

[ f'(x) = \lim_{h \to 0} \frac{-10x - 5h}{(x^2 + 7)((x + h)^2 + 7)} ]

[ f'(x) = \frac{-10x}{(x^2 + 7)^2} ]

Therefore, the derivative of ( f(x) = \frac{5}{x^2 + 7} ) using the difference quotient is ( f'(x) = \frac{-10x}{(x^2 + 7)^2} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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