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How do you differentiate #(x+x^(1/2))(cosx+x^(1/3))#?

Answer 1

Ezekiel Alexander

#f(x)' = (x + sqrt(x))(1/(3x^(2/3)) - sin x) + (cos x + root(3)(x))(1 + 1/(2sqrt(x)))#

Apply a combination of the Product and Power rules:

Product Rule: #(u*v)' = u*v' + v*u'# Power Rule: #(u^n)' = n*u^(n-1)*u'#

Let #u = x + x^(1/2)# so #u' = 1 + 1/2x^(-1/2)# Let #v = cos x + x^(1/3)# so #v' = -sin x + 1/3x^(-2/3)#

#f(x)' = (x + x^(1/2))( -sin x + 1/3x^(-2/3)) + (cos x + x^(1/3))(1 + 1/2x^(-1/2))#

So:

#f(x)' = (x + sqrt(x))(1/(3x^(2/3)) - sin x) + (cos x + root(3)(x))(1 + 1/(2sqrt(x)))#

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Answer 2

Adam Adkins

To differentiate ( (x+x^{1/2})(\cos x+x^{1/3}) ), you would use the product rule. Apply the product rule to the two functions, then differentiate each part separately, and combine them using the product rule formula. The result is the derivative of the given expression.

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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.