What is the d/(dy)(x^{2y}) ?

The d/(dy)(x^{2y}) is 2x^{2y}ln(x)

What is the first d/(dy)(x^{2y}) ?

The first d/(dy)(x^{2y}) is 2x^{2y}ln(x)

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\frac{d}{d y} (x^{2 y})

2 x^{2 y} ln (x)

Solution steps

\frac{d}{d y} (x^{2 y})

Treat

x

as a constant

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{2 y} = e^{2 y l n (x)}

= \frac{d}{d y} (e^{2 y l n (x)})

Apply the chain rule:

e^{2 y l n (x)} \frac{d}{d y} (2 y ln (x))

= e^{2 y l n (x)} \frac{d}{d y} (2 y ln (x))

\frac{d}{d y} (2 y ln (x)) = 2 ln (x)

= e^{2 y l n (x)} \cdot 2 ln (x)

e^{2 y l n (x)} = x^{2 y}

= 2 x^{2 y} ln (x)

36 y^{^{''}} + 36 y^{^{'}} + 11 y = 0, y (0) = 6, y^{^{'}} (0) = 5

\frac{6}{3 x + 1}, at a = - 1

\int (cos (x) + \frac{1}{8} x) d x

x \to 3 - lim (e^{\frac{2}{3 - x}})