Final answer:
The derivative of the function y = (5x^6 - 3x^4 + 2x^2 - 1)(4x^7 + 3x^5 - 5x^2 + 4x) is found using the product rule by first finding the derivatives of the individual factors and then applying (u*v)' = u'v + uv' to get the result without simplifying.
Step-by-step explanation:
To use the product rule for computing the derivative of the function y = (5x^6 - 3x^4 + 2x^2 - 1)(4x^7 + 3x^5 - 5x^2 + 4x), you need to designate each factor as a separate function. Let the first factor be u(x) = 5x^6 - 3x^4 + 2x^2 - 1 and the second factor be v(x) = 4x^7 + 3x^5 - 5x^2 + 4x. The product rule can be stated as (u*v)' = u'v + uv'. First, differentiate each factor separately.
The derivatives of the individual factors with respect to x are:
u'(x) = 30x^5 - 12x^3 + 4x
v'(x) = 28x^6 + 15x^4 - 10x + 4
Next, apply the product rule:
y' = (5x^6 - 3x^4 + 2x^2 - 1)(28x^6 + 15x^4 - 10x + 4) + (4x^7 + 3x^5 - 5x^2 + 4x)(30x^5 - 12x^3 + 4x)
This result is the derivative of the given function without simplification, applying the power rule of differentiation where necessary.