How do you differentiate #f(x)=-xe^x*(4-x)/6# using the product rule?
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To differentiate the function ( f(x) = -\frac{x e^x (4-x)}{6} ) using the product rule, you can follow these steps:
- Identify the two functions being multiplied: ( u(x) = -\frac{x(4-x)}{6} ) and ( v(x) = e^x ).
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Differentiate ( u(x) ) and ( v(x) ) separately.
- Substitute the derivatives and the original functions into the product rule formula.
Here's how to do it step by step:
- ( u(x) = -\frac{x(4-x)}{6} ) and ( v(x) = e^x ).
- ( u'(x) = \frac{d}{dx}(-\frac{x(4-x)}{6}) ) and ( v'(x) = \frac{d}{dx}(e^x) ).
- ( u'(x) = -\frac{1}{6} (4 - 2x) ) and ( v'(x) = e^x ).
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Substitute the derivatives and the original functions: ( f'(x) = -\frac{1}{6}(4 - 2x)e^x + (-\frac{x(4-x)}{6})e^x ).
- Simplify the expression: ( f'(x) = -\frac{1}{6}(4e^x - 2xe^x) -\frac{x(4-x)}{6}e^x ).
- Further simplify: ( f'(x) = -\frac{4e^x}{6} + \frac{2xe^x}{6} -\frac{x(4-x)e^x}{6} ).
- Combine like terms: ( f'(x) = -\frac{2e^x}{3} + \frac{xe^x}{3} -\frac{x(4-x)e^x}{6} ).
Thus, the derivative of ( f(x) ) is ( f'(x) = -\frac{2e^x}{3} + \frac{xe^x}{3} -\frac{x(4-x)e^x}{6} ).
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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.
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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