Differentiate y=(4x+x^(1/5))^(-1/5)

Set u as composite function: We need to differentiate the function y=(4x+x^(1/5))^(-1/5) with respect to x. We will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.Find dy/du: Let's set u = 4x + x^(1/5) so that y = u^(-1/5). We will first find the derivative of y with respect to u, which is dy/du = (-1/5) * u^(-6/5).Find du/dx: Next, we need to find the derivative of u with respect to x, which is du/dx = d/dx (4x + x^(1/5)). We will differentiate each term separately. The derivative of 4x with respect to x is 4, and the derivative of x^(1/5) with respect to x is (1/5)x^(-4/5).Apply chain rule: Now we have du/dx = 4 + (1/5)x^(-4/5). We can simplify this to du/dx = 4 + (1/5)x^(-4/5).Substitute derivatives: Using the chain rule, the derivative of y with respect to x is dy/dx = (dy/du) * (du/dx). We substitute the derivatives we found into this formula to get dy/dx = (-1/5) * u^(-6/5) * (4 + (1/5)x^(-4/5)).Simplify expression: We substitute back u = 4x + x^(1/5) into the derivative to get dy/dx = (-1/5) * (4x + x^(1/5))^(-6/5) * (4 + (1/5)x^(-4/5)).Final derivative: The final step is to simplify the expression if possible. However, in this case, the expression is already in its simplest form, so the derivative of y with respect to x is dy/dx = (-1/5) * (4x + x^(1/5))^(-6/5) * (4 + (1/5)x^(-4/5)).

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Differentiate $y=(4x+x^{\frac{1}{5}})^{-\frac{1}{5}}$

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Q. Differentiate

y=(4x+x^{\frac{1}{5}})^{-\frac{1}{5}}

Set $u$ as composite function: We need to differentiate the function $y=(4x+x^{1/5})^{-1/5}$ with respect to $x$ . We will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Find $\frac{dy}{du}$ : Let's set $u = 4x + x^{\frac{1}{5}}$ so that $y = u^{-\frac{1}{5}}$ . We will first find the derivative of $y$ with respect to $u$ , which is $\frac{dy}{du} = (-\frac{1}{5}) \cdot u^{-\frac{6}{5}}$ .
Find $\frac{du}{dx}$ : Next, we need to find the derivative of $u$ with respect to $x$ , which is $\frac{du}{dx} = \frac{d}{dx} (4x + x^{\frac{1}{5}})$ . We will differentiate each term separately. The derivative of $4x$ with respect to $x$ is $4$ , and the derivative of $x^{\frac{1}{5}}$ with respect to $x$ is $\left(\frac{1}{5}\right)x^{-\frac{4}{5}}$ .
Apply chain rule: Now we have $\frac{du}{dx} = 4 + \left(\frac{1}{5}\right)x^{-\frac{4}{5}}$ . We can simplify this to $\frac{du}{dx} = 4 + \left(\frac{1}{5}\right)x^{-\frac{4}{5}}$ .
Substitute derivatives: Using the chain rule, the derivative of $y$ with respect to $x$ is $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ . We substitute the derivatives we found into this formula to get $\frac{dy}{dx} = \left(-\frac{1}{5}\right) \cdot u^{-\frac{6}{5}} \cdot \left(4 + \left(\frac{1}{5}\right)x^{-\frac{4}{5}}\right)$ .
Simplify expression: We substitute back $u = 4x + x^{1/5}$ into the derivative to get $\frac{dy}{dx} = \left(-\frac{1}{5}\right) \cdot \left(4x + x^{1/5}\right)^{-6/5} \cdot \left(4 + \left(\frac{1}{5}\right)x^{-4/5}\right)$ .
Final derivative: The final step is to simplify the expression if possible. However, in this case, the expression is already in its simplest form, so the derivative of $y$ with respect to $x$ is $\frac{dy}{dx} = \left(-\frac{1}{5}\right) \cdot \left(4x + x^{\frac{1}{5}}\right)^{-\frac{6}{5}} \cdot \left(4 + \frac{1}{5}x^{-\frac{4}{5}}\right)$ .