# How do you use the Product Rule to find the derivative of #6e^{9x} sin(6x)#?

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To find the derivative of (6e^{9x} \sin(6x)) using the Product Rule, we differentiate each term separately and then apply the rule. The Product Rule states that if (u(x)) and (v(x)) are differentiable functions of (x), then the derivative of their product (u(x)v(x)) is given by (u'(x)v(x) + u(x)v'(x)).

Let (u(x) = 6e^{9x}) and (v(x) = \sin(6x)).

Then, (u'(x) = 6 \cdot 9e^{9x} = 54e^{9x}) (using the chain rule for the exponential function) and (v'(x) = 6\cos(6x)) (using the derivative of sine function).

Now, applying the Product Rule, we have:

[\frac{d}{dx} (u(x)v(x)) = u'(x)v(x) + u(x)v'(x)]

[\frac{d}{dx} (6e^{9x} \sin(6x)) = (54e^{9x})\sin(6x) + 6e^{9x}\cos(6x)]

So, the derivative of (6e^{9x} \sin(6x)) with respect to (x) is (54e^{9x}\sin(6x) + 6e^{9x}\cos(6x)).

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