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How do you differentiate #g(z) = z^2cos(3z)e^(2z)# using the product rule?

Answer 1

Isaiah Allen

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Answer 2

Elijah Abram

To differentiate ( g(z) = z^2 \cos(3z) e^{2z} ) using the product rule:

Identify the two functions being multiplied: ( u(z) = z^2 \cos(3z) ) and ( v(z) = e^{2z} ).
Apply the product rule formula: ( (uv)' = u'v + uv' ).
Find the derivatives of ( u(z) ) and ( v(z) ) separately.
- Derivative of ( u(z) ): ( u'(z) = (2z \cos(3z) - 3z^2 \sin(3z)) e^{2z} ).
- Derivative of ( v(z) ): ( v'(z) = 2e^{2z} ).
Substitute the derivatives and original functions into the product rule formula.
- ( g'(z) = (2z \cos(3z) - 3z^2 \sin(3z)) e^{2z} \cdot e^{2z} + z^2 \cos(3z) \cdot 2e^{2z} ).
Simplify the expression if necessary.

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Answer from HIX Tutor

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.