How do you find the derivative of the function #g(t)=(4t)/(t+1)#?
So, applying this to the function at hand, we see that:
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To find the derivative of the function ( g(t) = \frac{4t}{t+1} ), you can use the quotient rule. The quotient rule states that if you have a function in the form ( \frac{u}{v} ), where ( u ) and ( v ) are functions of ( t ), then the derivative is given by:
[ \frac{d}{dt} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dt} - u \frac{dv}{dt}}{v^2} ]
In this case, ( u(t) = 4t ) and ( v(t) = t + 1 ).
Now, differentiate ( u ) and ( v ) with respect to ( t ): [ \frac{du}{dt} = 4 ] [ \frac{dv}{dt} = 1 ]
Now apply the quotient rule: [ \frac{d}{dt} \left( \frac{4t}{t+1} \right) = \frac{(t+1)(4) - (4t)(1)}{(t+1)^2} ]
Simplify the expression: [ \frac{(4t + 4) - (4t)}{(t+1)^2} = \frac{4}{(t+1)^2} ]
So, the derivative of the function ( g(t) ) is ( \frac{4}{(t+1)^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.
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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