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

Problem

d()/d(x)*(2*x−5)4*(x2+x+1)5

Solution

1. Identify the product rule for differentiation, which states 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 to the expression where ƒ(x)=(2*x−5)4 and g(x)=(x2+x+1)5

d()/d(x)*(2*x−5)4*(x2+x+1)5=(2*x−5)4d(x2+x+1)/d(x)+(x2+x+1)5d(2*x−5)/d(x)

3. Apply the chain rule to differentiate the individual components.

d(x2+x+1)/d(x)=5*(x2+x+1)4*(2*x+1)

d(2*x−5)/d(x)=4*(2*x−5)3*(2)=8*(2*x−5)3

4. Substitute these derivatives back into the product rule expression.

(2*x−5)4⋅5*(x2+x+1)4*(2*x+1)+(x2+x+1)5⋅8*(2*x−5)3

5. Factor out the greatest common factors, which are (2*x−5)3 and (x2+x+1)4

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

6. Simplify the expression inside the brackets by expanding the terms.

5*(4*x2−8*x−5)+8*x2+8*x+8

20*x2−40*x−25+8*x2+8*x+8

28*x2−32*x−17

Final Answer

d()/d(x)*(2*x−5)4*(x2+x+1)5=(2*x−5)3*(x2+x+1)4*(28*x2−32*x−17)

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