Find the Derivative - d/dx y=(x^6)/(1-x^5)

Problem

d()/d(x)(x6)/(1−x5)

Solution

1. Identify the function as a quotient of two terms, u=x6 and v=1−x5 which requires the quotient rule.

2. Recall the quotient rule formula for a function u/v

d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator u=x6 with respect to x

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

4. Differentiate the denominator v=1−x5 with respect to x

d(1−x5)/d(x)=−5*x4

5. Substitute these components into the quotient rule formula.

d(y)/d(x)=((1−x5)*(6*x5)−(x6)*(−5*x4))/((1−x5)2)

6. Distribute the terms in the numerator.

d(y)/d(x)=(6*x5−6*x10+5*x10)/((1−x5)2)

7. Combine like terms in the numerator to find the final simplified expression.

d(y)/d(x)=(6*x5−x10)/((1−x5)2)

Final Answer

d()/d(x)(x6)/(1−x5)=(6*x5−x10)/((1−x5)2)

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