question 12 of 15, step 1 of 1 given the equation x - y^5 = 3, find dx/dy by implicit differentiation.

Explanation:

Step1: Differentiate both sides

Differentiate $x - y^{5}=3$ with respect to $y$. The derivative of $x$ with respect to $y$ is $\frac{dx}{dy}$, and the derivative of $y^{5}$ with respect to $y$ is $5y^{4}$ (using the power - rule $\frac{d}{dy}(y^{n})=ny^{n - 1}$), and the derivative of the constant 3 with respect to $y$ is 0. So we have $\frac{dx}{dy}-5y^{4}=0$.

Step2: Solve for $\frac{dx}{dy}$

Add $5y^{4}$ to both sides of the equation $\frac{dx}{dy}-5y^{4}=0$. We get $\frac{dx}{dy}=5y^{4}$.

Answer:

$\frac{dx}{dy}=5y^{4}$