Differentiate implicitly to find $\frac{d y}{d x}$. \[ y^{6}=x^{5} \] \[ \frac{d y}{d x}= \]
Solution
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 \(6y^{5} \frac{dy}{dx}\).
Step 3 ：Differentiating \(x^{5}\) with respect to \(x\) gives \(5x^{4}\).
Step 4 ：Setting these two derivatives equal to each other gives the equation \(6y^{5} \frac{dy}{dx} = 5x^{4}\).
Step 5 ：Solving for \(\frac{dy}{dx}\) gives \(\frac{dy}{dx} = \frac{5x^{4}}{6y^{5}}\).
Step 6 ：Substituting \(y^{6}=x^{5}\) into the equation gives \(\frac{dy}{dx} = \frac{5x^{4}}{6x^{5/6}}\).
Step 7 ：Simplifying the equation gives \(\frac{dy}{dx} = \frac{5}{6}x^{4-5/6}\).
Step 8 ：Finally, simplifying the exponent gives \(\frac{dy}{dx} = \frac{5}{6}x^{19/6}\).
Step 9 ：So, the derivative of \(y\) with respect to \(x\) is \(\boxed{\frac{5}{6}x^{19/6}}\).
