Find dy/dx y=(x^-2+x)^-3

Problem

d()/d(x)*(x(−2)+x)(−3)

Solution

1. Identify the outer and inner functions to apply the Chain Rule, where the outer function is u(−3) and the inner function is u=x(−2)+x

2. Apply the Power Rule to the outer function by bringing the exponent down and subtracting one from the exponent.

d(y)/d(x)=−3*(x(−2)+x)(−4)⋅d(x(−2)+x)/d(x)

3. Differentiate the inner expression x(−2)+x using the Power Rule for each term.

d(x(−2)+x)/d(x)=−2*x(−3)+1

4. Substitute the derivative of the inner function back into the expression.

d(y)/d(x)=−3*(x(−2)+x)(−4)*(−2*x(−3)+1)

5. Simplify the expression by distributing the −3 into the second set of parentheses.

d(y)/d(x)=(6*x(−3)−3)*(x(−2)+x)(−4)

Final Answer

d(y)/d(x)=(6*x(−3)−3)/((x(−2)+x)4)

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