# How do you differentiate #y=e^(-5x)cos3x#?

Use the Product Rule

The product rule is:

Substituting into the product rule:

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To differentiate ( y = e^{-5x}\cos(3x) ), you would use the product rule. The product rule states that if you have two functions, ( u(x) ) and ( v(x) ), then the derivative of their product, ( u(x) \cdot v(x) ), is given by ( u'(x) \cdot v(x) + u(x) \cdot v'(x) ).

In this case, ( u(x) = e^{-5x} ) and ( v(x) = \cos(3x) ).

The derivatives of ( e^{-5x} ) and ( \cos(3x) ) with respect to ( x ) are ( -5e^{-5x} ) and ( -3\sin(3x) ), respectively.

Therefore, applying the product rule:

[ \frac{d}{dx}(e^{-5x}\cos(3x)) = -5e^{-5x}\cos(3x) + e^{-5x}(-3\sin(3x)) ]

[ = -5e^{-5x}\cos(3x) - 3e^{-5x}\sin(3x) ]

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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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