Find the Derivative - d/dx y=(x^2+2)^2(x^4+4)^4

Problem

d()/d(x)*(x2+2)2*(x4+4)4

Solution

1. Identify the rule needed for the derivative of a product of two functions, which is the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions u=(x2+2)2 and v=(x4+4)4

3. Apply the chain rule to find d(u)/d(x)

d(x2+2)/d(x)=2*(x2+2)⋅2*x=4*x*(x2+2)

4. Apply the chain rule to find d(v)/d(x)

d(x4+4)/d(x)=4*(x4+4)3⋅4*x3=16*x3*(x4+4)3

5. Substitute these into the product rule formula:

d(y)/d(x)=(x2+2)2⋅16*x3*(x4+4)3+(x4+4)4⋅4*x*(x2+2)

6. Factor out the greatest common factor, which is 4*x*(x2+2)*(x4+4)3

d(y)/d(x)=4*x*(x2+2)*(x4+4)3*[4*x2*(x2+2)+(x4+4)]

7. Simplify the expression inside the brackets:

4*x4+8*x2+x4+4=5*x4+8*x2+4

8. Combine all terms to reach the final derivative.

Final Answer

d()/d(x)*(x2+2)2*(x4+4)4=4*x*(x2+2)*(x4+4)3*(5*x4+8*x2+4)

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