Identify Function: We are asked to find the derivative of the function x^(4)y + x^(2)y^(4) with respect to x. This is a partial differentiation problem because the function has two variables, x and y. We will treat y as a constant when differentiating with respect to x.Differentiate x^(4)y: First, we differentiate the term x^(4)y with respect to x. Using the power rule, the derivative of x^n with respect to x is nx^(n-1). Since y is treated as a constant, it behaves like a coefficient. (d/dx)(x^(4)y) = y * (d/dx)(x^(4)) = y * 4x^(4-1) = 4yx^(3).Differentiate x^(2)y^(4): Next, we differentiate the term x^(2)y^(4) with respect to x. Again, using the power rule and treating y^(4) as a constant, we get: (d/dx)(x^(2)y^(4)) = y^(4) * (d/dx)(x^(2)) = y^(4) * 2x^(2-1) = 2y^(4)x.Combine Derivatives: Now, we combine the derivatives of both terms to find the derivative of the entire function with respect to x. (d/dx)(x^(4)y + x^(2)y^(4)) = (d/dx)(x^(4)y) + (d/dx)(x^(2)y^(4)) = 4yx^(3) + 2y^(4)x.Final Result: We have found the derivative of the function x^(4)y + x^(2)y^(4) with respect to x without any mathematical errors.

Calculus

Find derivatives of using multiple formulae

Full solution

x^{4} y+x^{2} y^{4}

Identify Function: We are asked to find the derivative of the function $x^{4}y + x^{2}y^{4}$ with respect to $x$ . This is a partial differentiation problem because the function has two variables, $x$ and $y$ . We will treat $y$ as a constant when differentiating with respect to $x$ .
Differentiate $x^{4}y$ : First, we differentiate the term $x^{4}y$ with respect to $x$ . Using the power rule, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$ . Since $y$ is treated as a constant, it behaves like a coefficient. $\newline$ $(\frac{d}{dx})(x^{4}y) = y \cdot (\frac{d}{dx})(x^{4}) = y \cdot 4x^{4-1} = 4yx^{3}$ .
Differentiate $x^{2}y^{4}$ : Next, we differentiate the term $x^{2}y^{4}$ with respect to $x$ . Again, using the power rule and treating $y^{4}$ as a constant, we get: $\newline$ $(\frac{d}{dx})(x^{2}y^{4}) = y^{4} \cdot (\frac{d}{dx})(x^{2}) = y^{4} \cdot 2x^{2-1} = 2y^{4}x$ .
Combine Derivatives: Now, we combine the derivatives of both terms to find the derivative of the entire function with respect to $x$ . $(\frac{d}{dx})(x^{4}y + x^{2}y^{4}) = (\frac{d}{dx})(x^{4}y) + (\frac{d}{dx})(x^{2}y^{4}) = 4yx^{3} + 2y^{4}x.$
Final Result: We have found the derivative of the function $x^{4}y + x^{2}y^{4}$ with respect to $x$ without any mathematical errors.