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

Problem

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

5. Combine the parts using the product rule formula:

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

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

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

7. Simplify the expression inside the brackets:

8*x2+8*x+8+5*(4*x2−4*x−3)

8*x2+8*x+8+20*x2−20*x−15

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!