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

Problem

d()/d(x)*(2*x−3)4*(x2+x+1)5

Solution

1. Identify the product rule for differentiation, which states that d()/d(x)*[u(x)*v(x)]=u(x)′*v(x)+u(x)*v(x)′

2. Assign the functions u(x)=(2*x−3)4 and v(x)=(x2+x+1)5

3. Apply the chain rule to find u(x)′=4*(2*x−3)3⋅d(2*x−3)/d(x)=4*(2*x−3)3*(2)=8*(2*x−3)3

4. Apply the chain rule to find v(x)′=5*(x2+x+1)4⋅d(x2+x+1)/d(x)=5*(x2+x+1)4*(2*x+1)

5. Substitute these derivatives into the product rule formula.

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

6. Factor out the greatest common factor, which is (2*x−3)3*(x2+x+1)4

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

7. Expand the terms inside the brackets.

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

8. Distribute the 5 and combine like terms.

ƒ(x)′=(2*x−3)3*(x2+x+1)4*[8*x2+8*x+8+20*x2−20*x−15]

ƒ(x)′=(2*x−3)3*(x2+x+1)4*(28*x2−12*x−7)

Final Answer

d()/d(x)*(2*x−3)4*(x2+x+1)5=(2*x−3)3*(x2+x+1)4*(28*x2−12*x−7)

Want more problems? Check here!