Find the Derivative - d/dy (1/(y^2)-5/(y^4))(y+7y^3)

Problem

d()/d(y)*(1/(y2)−5/(y4))*(y+7*y3)

Solution

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

(1/(y2)−5/(y4))*(y+7*y3)=y/(y2)+(7*y3)/(y2)−(5*y)/(y4)−(35*y3)/(y4)

2. Simplify each term using the laws of exponents.

1/y+7*y−5/(y3)−35/y

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

7*y−5/(y3)−34/y

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

7*y−5*y(−3)−34*y(−1)

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

d()/d(y)*(7*y−5*y(−3)−34*y(−1))=7*(1)*y0−5*(−3)*y(−4)−34*(−1)*y(−2)

6. Simplify the resulting coefficients and exponents.

7+15*y(−4)+34*y(−2)

7. Convert the negative exponents back into fraction form for the final result.

7+15/(y4)+34/(y2)

Final Answer

d()/d(y)*(1/(y2)−5/(y4))*(y+7*y3)=7+34/(y2)+15/(y4)

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