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How do you differentiate #f(x)=(1/x)e^xtanx-x*cosx# using the product rule?

Answer 1

Charles Arocha

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Answer 2

Emma Aldridge

To differentiate the function ( f(x) = \frac{1}{x} \cdot e^x \cdot \tan(x) - x \cdot \cos(x) ) using the product rule, follow these steps:

Identify the two functions being multiplied together: ( u(x) = \frac{1}{x} \cdot e^x \cdot \tan(x) ) and ( v(x) = -x \cdot \cos(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 back into the product rule formula.

Applying the product rule: [ f'(x) = \left(\frac{-1}{x^2} \cdot e^x \cdot \tan(x) + \frac{1}{x} \cdot e^x \cdot \sec^2(x) \cdot \tan(x) + \frac{1}{x} \cdot e^x \cdot \sec^2(x) \cdot \tan(x) - \cos(x)\right) - \left(\cos(x) - x \cdot \sin(x)\right) ]

Simplify the expression to obtain the final result for ( f'(x) ).

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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.

How do you differentiate #f(x)=(1/x)*e^x*tanx-x*cosx# using the product rule?

How do you differentiate #f(x)=(1/x)e^xtanx-x*cosx# using the product rule?