Differentiate implicitly to find $\frac{d y}{d x}$ .
$y^{6} = x^{5}$
$\frac{d y}{d x} =$

Answer

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Answer

So, the derivative of $y$ with respect to $x$ is $\frac{5}{6} x^{19 / 6}$ .

Steps

Step 1 ：Given the equation $y^{6} = x^{5}$ , we can differentiate both sides with respect to $x$ .

Step 2 ：Differentiating $y^{6}$ with respect to $x$ gives $6 y^{5} \frac{d y}{d x}$ .

Step 3 ：Differentiating $x^{5}$ with respect to $x$ gives $5 x^{4}$ .

Step 4 ：Setting these two derivatives equal to each other gives the equation $6 y^{5} \frac{d y}{d x} = 5 x^{4}$ .

Step 5 ：Solving for $\frac{d y}{d x}$ gives $\frac{d y}{d x} = \frac{5 x^{4}}{6 y^{5}}$ .

Step 6 ：Substituting $y^{6} = x^{5}$ into the equation gives $\frac{d y}{d x} = \frac{5 x^{4}}{6 x^{5 / 6}}$ .

Step 7 ：Simplifying the equation gives $\frac{d y}{d x} = \frac{5}{6} x^{4 - 5 / 6}$ .

Step 8 ：Finally, simplifying the exponent gives $\frac{d y}{d x} = \frac{5}{6} x^{19 / 6}$ .

Step 9 ：So, the derivative of $y$ with respect to $x$ is $\frac{5}{6} x^{19 / 6}$ .