Find dy/dx 16e^y=x^4-y^3

Problem

16*ey=x4−y3

Solution

1. Differentiate both sides with respect to x using implicit differentiation.

2. Apply the chain rule to the terms involving y treating y as a function of x

(d(16)*ey)/d(x)=d(x4−y3)/d(x)

3. Compute the derivatives for each term.

16*eyd(y)/d(x)=4*x3−3*y2d(y)/d(x)

4. Group the terms containing d(y)/d(x) on one side of the equation.

16*eyd(y)/d(x)+3*y2d(y)/d(x)=4*x3

5. Factor out d(y)/d(x) from the left side.

d(y)/d(x)*(16*ey+3*y2)=4*x3

6. Solve for dy/dx by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(4*x3)/(16*ey+3*y2)

Final Answer

d(y)/d(x)=(4*x3)/(16*ey+3*y2)

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