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How do you differentiate # f(x)=(2x+1)^(5/2) (4x-1)^(3/4) # using the product rule?

Answer 1

Evelyn Agard

See pointer for calculation method in Explanation

Let #u= (2x+1)^(5/2)" "# then we have #(du)/dx = 5/2(2x+1)^(3/2)(2)#

Let #v = (4x-1)^(3/4)" "# then #(dv)/dx = 3/4(4x-1)^(-1/4)(4)#

So #(du)/dx = 5/(cancel(2))(2x+1)^(3/2)cancel((2))#

#(dv)/dx = 3/(cancel(4))(4x-1)^(-1/4)(cancel(4)) = 3/((4x-1)^(1/4))#

Substitute into:

#f^'(x) = v(du)/dx + u(dv)/dx#

Once that is done it is a matter of simplification!

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Answer 2

Scarlett Allison

To differentiate ( f(x) = (2x + 1)^{5/2} \cdot (4x - 1)^{3/4} ) using the product rule, follow these steps:

Identify the two functions being multiplied: ( u = (2x + 1)^{5/2} ) and ( v = (4x - 1)^{3/4} ).
Calculate the derivatives of each function: ( u' = \frac{5}{2}(2x + 1)^{3/2} \cdot 2 ) and ( v' = \frac{3}{4}(4x - 1)^{-1/4} \cdot 4 ).
Apply the product rule: ( f'(x) = u'v + uv' ).

After substituting the derivatives and the original functions into the product rule formula, simplify the expression to obtain the derivative of ( f(x) ).

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Answer from HIX Tutor

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