Find the Derivative - d/dy (1/(y^2)-3/(y^4))(y+5y^3)

Problem

d()/d(y)*(1/(y2)−3/(y4))*(y+5*y3)

Solution

1. Expand the expression by multiplying the two binomials to simplify the differentiation process.

(1/(y2)−3/(y4))*(y+5*y3)=y/(y2)+(5*y3)/(y2)−(3*y)/(y4)−(15*y3)/(y4)

2. Simplify each term using the laws of exponents.

1/y+5*y−3/(y3)−15/y

3. Combine like terms to reduce the number of terms to differentiate.

5*y−14/y−3/(y3)

4. Rewrite the expression using negative exponents to prepare for the power rule.

5*y−14*y(−1)−3*y(−3)

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

d()/d(y)*(5*y−14*y(−1)−3*y(−3))=5*(1)*y0−14*(−1)*y(−2)−3*(−3)*y(−4)

6. Simplify the resulting expression.

5+14*y(−2)+9*y(−4)

7. Convert back to fraction notation for the final result.

5+14/(y2)+9/(y4)

Final Answer

d()/d(y)*(1/(y2)−3/(y4))*(y+5*y3)=5+14/(y2)+9/(y4)

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