Find the Derivative - d/dx (2x-3)^4(x^2+x+1)^5

Problem

d()/d(x)*(2*x−3)^4*(x^2+x+1)^5

Solution

1. Identify the rule needed for the product of two functions, which is the product rule: d()/d(x)*[ƒ(x)*g(x)]=ƒ(x)^′*g(x)+ƒ(x)*g(x)^′

2. Assign the functions where ƒ(x)=(2*x−3)^4 and g(x)=(x^2+x+1)^5

3. Apply the chain rule to find the derivative of ƒ(x)

d(2*x−3)/d(x)=4*(2*x−3)^3⋅2=8*(2*x−3)^3

4. Apply the chain rule to find the derivative of g(x)

d(x^2+x+1)/d(x)=5*(x^2+x+1)^4⋅(2*x+1)

5. Combine the parts using the product rule formula:

d()/d(x)*(2*x−3)^4*(x^2+x+1)^5=8*(2*x−3)^3*(x^2+x+1)^5+(2*x−3)^4⋅5*(x^2+x+1)^4*(2*x+1)

6. Factor out the greatest common factors, which are (2*x−3)^3 and (x^2+x+1)^4

(2*x−3)^3*(x^2+x+1)^4*[8*(x^2+x+1)+5*(2*x−3)*(2*x+1)]

7. Simplify the expression inside the brackets:

8*x^2+8*x+8+5*(4*x^2−4*x−3)

8*x^2+8*x+8+20*x^2−20*x−15

28*x^2−12*x−7

Final Answer

d()/d(x)*(2*x−3)^4*(x^2+x+1)^5=(2*x−3)^3*(x^2+x+1)^4*(28*x^2−12*x−7)

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