Find the Derivative - d/dy (1/(y^2)-9/(y^4))(y+3y^3)

Problem

d()/d(y)*(1/(y2)−9/(y4))*(y+3*y3)

Solution

1. Expand the expression by multiplying the two binomials using the distributive property.

(1/(y2)−9/(y4))*(y+3*y3)=y/(y2)+(3*y3)/(y2)−(9*y)/(y4)−(27*y3)/(y4)

2. Simplify each term by reducing the powers of y

1/y+3*y−9/(y3)−27/y

3. Combine like terms, specifically the terms involving 1/y

3*y−26/y−9/(y3)

4. Rewrite the expression using negative exponents to prepare for differentiation.

3*y−26*y(−1)−9*y(−3)

5. Apply the power rule, which states d(yn)/d(y)=n*y(n−1) to each term individually.

d()/d(y)*(3*y)−d()/d(y)*(26*y(−1))−d()/d(y)*(9*y(−3))

3−26*(−1)*y(−2)−9*(−3)*y(−4)

6. Simplify the coefficients and rewrite the expression with positive exponents.

3+26*y(−2)+27*y(−4)

3+26/(y2)+27/(y4)

Final Answer

d()/d(y)*(1/(y2)−9/(y4))*(y+3*y3)=3+26/(y2)+27/(y4)

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