# What is the derivative of #x(6^(-2x))#?

By Product Rule,

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To find the derivative of ( x \cdot 6^{-2x} ), you can use the product rule and the chain rule of differentiation. The derivative is:

[ \frac{d}{dx}(x \cdot 6^{-2x}) = x \cdot \frac{d}{dx}(6^{-2x}) + 6^{-2x} \cdot \frac{d}{dx}(x) ]

Applying the chain rule and the power rule, the derivative of ( 6^{-2x} ) is:

[ \frac{d}{dx}(6^{-2x}) = -2 \cdot 6^{-2x} \cdot \ln(6) ]

The derivative of ( x ) is simply ( 1 ).

Substituting these derivatives back into the original expression, you get:

[ \frac{d}{dx}(x \cdot 6^{-2x}) = x \cdot (-2 \cdot 6^{-2x} \cdot \ln(6)) + 6^{-2x} \cdot 1 ]

This can be simplified as:

[ \frac{d}{dx}(x \cdot 6^{-2x}) = -2x \cdot 6^{-2x} \cdot \ln(6) + 6^{-2x} ]

So, the derivative of ( x \cdot 6^{-2x} ) is ( -2x \cdot 6^{-2x} \cdot \ln(6) + 6^{-2x} ).

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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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