Find the Derivative - d/dx 8x(x^2+9)^3

Problem

d()/d(x)*8*x*(x2+9)3

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=8*x and v=(x2+9)3

2. Apply the product rule, which states d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the first part u=8*x to get d(u)/d(x)=8

4. Differentiate the second part v=(x2+9)3 using the chain rule to get d(v)/d(x)=3*(x2+9)2⋅2*x=6*x*(x2+9)2

5. Substitute these components back into the product rule formula.

8*x⋅6*x*(x2+9)2+(x2+9)3⋅8

6. Simplify the terms by multiplying the coefficients.

48*x2*(x2+9)2+8*(x2+9)3

7. Factor out the greatest common factor, which is 8*(x2+9)2

8*(x2+9)2*[6*x2+(x2+9)]

8. Combine the terms inside the brackets.

8*(x2+9)2*(7*x2+9)

Final Answer

(d(8)*x*(x2+9)3)/d(x)=8*(x2+9)2*(7*x2+9)

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