Find the Derivative - d/dx (2x-7)^4(x^2+x+1)^5

Problem

d()/d(x)*(2*x−7)4*(x2+x+1)5

Solution

1. Identify the product rule for differentiation, which states that d()/d(x)*ƒ(x)*g(x)=ƒ(x)d(g(x))/d(x)+g(x)d(ƒ(x))/d(x)

2. Apply the product rule by setting ƒ(x)=(2*x−7)4 and g(x)=(x2+x+1)5

3. Differentiate the first part using the chain rule: d(2*x−7)/d(x)=4*(2*x−7)3⋅2=8*(2*x−7)3

4. Differentiate the second part using the chain rule: d(x2+x+1)/d(x)=5*(x2+x+1)4⋅(2*x+1)

5. Combine the results into the product rule formula.

d()/d(x)*(2*x−7)4*(x2+x+1)5=(2*x−7)4⋅5*(x2+x+1)4*(2*x+1)+(x2+x+1)5⋅8*(2*x−7)3

6. Factor out the greatest common factors, which are (2*x−7)3 and (x2+x+1)4

(2*x−7)3*(x2+x+1)4*[5*(2*x−7)*(2*x+1)+8*(x2+x+1)]

7. Expand the expression inside the brackets.

5*(4*x2−12*x−7)+8*x2+8*x+8

20*x2−60*x−35+8*x2+8*x+8

8. Simplify the terms inside the brackets by combining like terms.

28*x2−52*x−27

Final Answer

d()/d(x)*(2*x−7)4*(x2+x+1)5=(2*x−7)3*(x2+x+1)4*(28*x2−52*x−27)

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