Differentiate y=(2x+1)^(5)(x^(3)-x+1)^(4)

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Differentiate  y=(2x+1)^(5)(x^(3)-x+1)^(4)

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Apply Product Rule: We need to differentiate the function y=(2x+1)^(5)(x^(3)-x+1)^(4). This requires the use of the product rule for differentiation, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.Find Derivatives: Let's denote the first function as u=(2x+1)^(5) and the second function as v=(x^(3)-x+1)^(4). We will need to find the derivatives u' and v' separately.Differentiate u: First, we differentiate u with respect to x. Using the chain rule, we get u' = 5*(2x+1)^(4)*2, since the derivative of (2x+1) with respect to x is 2.Differentiate v: Now, we differentiate v with respect to x. Again, using the chain rule, we get v' = 4*(x^(3)-x+1)^(3)*(3x^(2)-1), since the derivative of (x^(3)-x) with respect to x is 3x^(2)-1.Apply Product Rule: Applying the product rule, the derivative of y with respect to x is: y' = u'v + uv'    = (5*(2x+1)^(4)*2)*(x^(3)-x+1)^(4) + (2x+1)^(5)*(4*(x^(3)-x+1)^(3)*(3x^(2)-1))Simplify y': Now we simplify the expression for y': y' = 10*(2x+1)^(4)*(x^(3)-x+1)^(4) + (2x+1)^(5)*4*(x^(3)-x+1)^(3)*(3x^(2)-1)Combine Like Terms: The final step is to combine like terms if possible, but in this case, the terms are not like terms and cannot be combined further. Therefore, the simplified derivative of y is: y' = 10*(2x+1)^(4)*(x^(3)-x+1)^(4) + 12*(2x+1)^(5)*(x^(3)-x+1)^(3)*(x^(2)-1/3)

Differentiate $y=(2x+1)^{5}(x^{3}-x+1)^{4}$

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Differentiate y=(2x+1)5(x3−x+1)4y=(2x+1)^{5}(x^{3}-x+1)^{4}y=(2x+1)5(x3−x+1)4

Differentiate $y=(2x+1)^{5}(x^{3}-x+1)^{4}$