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

Se the answer below:

By signing up, you agree to our Terms of Service and Privacy Policy

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) ).

By signing up, you agree to our Terms of Service and Privacy Policy

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7