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

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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

\int (3 x^{2} + 10 x) (x^{3} + 5 x^{2} + 6) d x

\frac{d y}{d x} = (y - x + 1)^{2} + 1

x \to 0 lim (\frac{0}{x ^{3}})

y^{^{'}} = 3 - \frac{y}{1000}

y = 49.99 sin (\frac{x}{5.0073}) + 49.99