Find dy/dx 15x=15y+5y^3+3y^5

Problem

15*x=15*y+5*y3+3*y5

Solution

1. Differentiate both sides with respect to x using the chain rule for terms involving y

2. Apply the power rule to each term on both sides of the equation.

(d(15)*x)/d(x)=d()/d(x)*(15*y+5*y3+3*y5)

3. Compute the derivatives on the left and right sides, noting that d(yn)/d(x)=n*y(n−1)d(y)/d(x)

15=15d(y)/d(x)+15*y2d(y)/d(x)+15*y4d(y)/d(x)

4. Factor out the common term d(y)/d(x) from the right side of the equation.

15=d(y)/d(x)*(15+15*y2+15*y4)

5. Divide both sides by the expression in the parentheses to isolate d(y)/d(x)

d(y)/d(x)=15/(15+15*y2+15*y4)

6. Simplify the fraction by dividing the numerator and the denominator by 15

d(y)/d(x)=1/(1+y2+y4)

Final Answer

d(y)/d(x)=1/(1+y2+y4)

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