Find the Derivative - d/dx [(x^3+x)(x^3-x)]

Problem

d()/d(x)*[(x3+x)*(x3−x)]

Solution

1. Identify the expression as a product of a sum and a difference, which follows the pattern (a+b)*(a−b)=a2−b2

2. Expand the expression using the difference of squares to simplify the function before differentiating.

(x3+x)*(x3−x)=(x3)2−(x)2

3. Simplify the powers using the rule (xa)b=x(a*b)

(x3)2−x2=x6−x2

4. Apply the power rule for differentiation, which states d(xn)/d(x)=n*x(n−1) to each term.

d(x6)/d(x)=6*x5

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

5. Combine the results to find the final derivative.

d(x6−x2)/d(x)=6*x5−2*x

Final Answer

d()/d(x)*[(x3+x)*(x3−x)]=6*x5−2*x

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