differentiate implicitly to find \\( \\frac { d y } { d x } \\).\n\n\\( y ^ { 9 } = x ^ { 4 } \\)\n\n\\(…

differentiate implicitly to find \\( \\frac { d y } { d x } \\).\n\n\\( y ^ { 9 } = x ^ { 4 } \\)\n\n\\( \\frac { d y } { d x } = \\)

differentiate implicitly to find \\( \\frac { d y } { d x } \\).\n\n\\( y ^ { 9 } = x ^ { 4 } \\)\n\n\\( \\frac { d y } { d x } = \\)

Answer

Explanation:

Step1: Differentiate both sides

Differentiate (y^{9}) and (x^{4}) with respect to (x). Using the chain rule ((u^n)^\prime = nu^{n - 1}u^\prime) for (y^{9}) (where (u = y)) and the power rule ((x^n)^\prime=nx^{n - 1}) for (x^{4}). We get (9y^{8}\frac{dy}{dx}=4x^{3}).

Step2: Solve for (\frac{dy}{dx})

Divide both sides of the equation (9y^{8}\frac{dy}{dx}=4x^{3}) by (9y^{8}). So (\frac{dy}{dx}=\frac{4x^{3}}{9y^{8}}).

Answer:

(\frac{4x^{3}}{9y^{8}})