# How do you use the product Rule to find the derivative of #g(t) = (3^t) - (4^t3)#?

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To use the product rule to find the derivative of ( g(t) = 3^t - 4^{t^3} ):

Let ( f(t) = 3^t ) and ( h(t) = 4^{t^3} ).

Now, apply the product rule:

( g'(t) = f'(t) \cdot h(t) + f(t) \cdot h'(t) ).

Find the derivatives of ( f(t) ) and ( h(t) ):

( f'(t) = (\ln(3) \cdot 3^t) ) and ( h'(t) = (\ln(4) \cdot 3t^2 \cdot 4^{t^3}) ).

Substitute these derivatives into the product rule:

( g'(t) = (\ln(3) \cdot 3^t) \cdot (4^{t^3}) + (3^t) \cdot (\ln(4) \cdot 3t^2 \cdot 4^{t^3}) ).

So, the derivative of ( g(t) = 3^t - 4^{t^3} ) using the product rule is:

( g'(t) = (\ln(3) \cdot 3^t) \cdot (4^{t^3}) + (3^t) \cdot (\ln(4) \cdot 3t^2 \cdot 4^{t^3}) ).

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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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