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How do do you differentiate #f(x)= (x+sinx)/(cosx)#?

Answer 1

Isaiah Allen

#f'(x) = frac{cos x + x sin x + 1}{cos^2 x}#

#f'(x) = frac{(1 + cos x) cos x - (x + sin x)(- sin x)}{cos^2 x}#

# = frac{cos x + cos^2 x + x sin x + sin^2 x}{cos^2 x}#

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Answer 2

Paisley Johnson

To differentiate ( f(x) = \frac{x + \sin(x)}{\cos(x)} ):

Apply the quotient rule, which states that if ( f(x) = \frac{u(x)}{v(x)} ), then ( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} ).
Identify ( u(x) = x + \sin(x) ) and ( v(x) = \cos(x) ).
Find the derivatives ( u'(x) ) and ( v'(x) ).
- ( u'(x) = 1 + \cos(x) )
- ( v'(x) = -\sin(x) )
Substitute the derivatives and the original functions into the quotient rule formula.
- ( f'(x) = \frac{(1 + \cos(x))\cos(x) - (x + \sin(x))(-\sin(x))}{[\cos(x)]^2} )
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.