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Sagot :
Given:
The given function is:
[tex]y=(4x^3-5x^2)(3x^6+2x^5)[/tex]First method: Multiplying factors and differentiating after arranging like terms together.
[tex]\begin{gathered} y=(4x^3-5x^2)(3x^6+2x^5) \\ =4x^3(3x^6+2x^5)-5x^2(3x^6+2x^5) \\ =12x^9+8x^8-15x^8-10x^7 \\ =12x^9-7x^{8^{}}-10x^7 \end{gathered}[/tex]Now we will differentiate y with respect to x by basic rules:
[tex]y^{\prime}=12(9)x^{9-1}-7(8)x^{8-1}-10(7)x^{7-1}[/tex]Solving further,
[tex]y^{\prime}=108x^8-56x^7-70x^6[/tex](b) Second method: Apply product rule to find the derivative:
Again,
[tex]y=(4x^3-5x^2)(3x^6+2x^5)[/tex]The product rule states:
[tex](uv)^{\prime}=u^{\prime}v+v^{\prime}u[/tex]Where u and v are the two factors multiplied.
So here we have:
[tex]\begin{gathered} u=4x^3-5x^2 \\ v=3x^6+2x^5 \end{gathered}[/tex]Finding the derivatives:
[tex]\begin{gathered} u^{\prime}=4(3)x^{3-1}-5(2)x^{2-1} \\ =12x^2-10x \end{gathered}[/tex]Similarly,
[tex]\begin{gathered} v^{\prime}=3(6)x^{6-1}+2(5)x^{5-1} \\ =18x^5+10x^4 \end{gathered}[/tex]Now put the values in the product rule,
[tex]\begin{gathered} y^{\prime}=(uv)^{\prime} \\ =u^{\prime}v+v^{\prime}u \\ =(12x^2-10x)(3x^6+2x^5)+(18x^5+10x^4)(4x^3-5x^2) \end{gathered}[/tex]Simplifying further,
[tex]\begin{gathered} y^{\prime}=12x^2(3x^6+2x^5)-10x(3x^6+2x^5)+18x^5(4x^3-5x^2)+10x^4(4x^3-5x^2) \\ =36x^8+24x^7-30x^7-20x^6+72x^8-90x^7+40x^7-50x^6 \\ =108x^8-56x^7-70x^6 \end{gathered}[/tex]This is the derivative obtained.
From above two methods, we can see the derivative is same in both the cases.
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