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

Problem

d()/d(x)*[(x6+x)*(x6−x)]

Solution

1. Recognize the expression as a difference of squares in the form (a+b)*(a−b) where a=x6 and b=x

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

(x6+x)*(x6−x)=(x6)2−(x)2

(x6+x)*(x6−x)=x12−x2

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

d(x12)/d(x)=12*x11

d(x2)/d(x)=2*x

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

d()/d(x)*[x12−x2]=12*x11−2*x

Final Answer

d()/d(x)*[(x6+x)*(x6−x)]=12*x11−2*x

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