What is the derivative of #(t-2)^3 (t-6)#?
Try using the Product and Chain Rule:
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To find the derivative of the function (t-2)^3(t-6), you can use the product rule of differentiation. The product rule states that if you have two functions, f(t) and g(t), then the derivative of their product is given by the formula:
[ (f(t)g(t))' = f'(t)g(t) + f(t)g'(t) ]
In this case, let ( f(t) = (t-2)^3 ) and ( g(t) = (t-6) ). Now, differentiate each function with respect to ( t ):
[ f'(t) = 3(t-2)^2 \times \frac{d}{dt}(t-2) ] [ g'(t) = \frac{d}{dt}(t-6) ]
Now, calculate the derivatives:
[ \frac{d}{dt}(t-2) = 1 ] [ \frac{d}{dt}(t-6) = 1 ]
Substitute these into the product rule formula:
[ (t-2)^3(t-6)' = (3(t-2)^2 \times 1)(t-6) + (t-2)^3 \times 1 ]
Simplify this expression to find the derivative.
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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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