Find the Derivative - d/dx [(x^6+x)(x^6-x)]

Problem

d()/d(x)*[(x^6+x)*(x^6−x)]

Solution

1. Recognize the expression as a difference of squares in the form (a+b)*(a−b) where a=x^6 and b=x

2. Simplify the expression by multiplying the binomials before differentiating to avoid using the product rule.

(x^6+x)*(x^6−x)=(x^6)^2−(x)^2

(x^6+x)*(x^6−x)=x^12−x^2

3. Apply the power rule, which states d(x^n)/d(x)=n*x^(n−1) to each term of the simplified polynomial.

d(x^12)/d(x)=12*x^11

d(x^2)/d(x)=2*x

4. Combine the results using the sum/difference rule for derivatives.

d()/d(x)*[x^12−x^2]=12*x^11−2*x

Final Answer

d()/d(x)*[(x^6+x)*(x^6−x)]=12*x^11−2*x

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