What is the d/(dy)(x^{8y}) ?

The d/(dy)(x^{8y}) is 8x^{8y}ln(x)

What is the first d/(dy)(x^{8y}) ?

The first d/(dy)(x^{8y}) is 8x^{8y}ln(x)

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\frac{d}{d y} (x^{8 y})

8 x^{8 y} ln (x)

Solution steps

\frac{d}{d y} (x^{8 y})

Treat

x

as a constant

Apply exponent rule:

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

x^{8 y} = e^{8 y l n (x)}

= \frac{d}{d y} (e^{8 y l n (x)})

Apply the chain rule:

e^{8 y l n (x)} \frac{d}{d y} (8 y ln (x))

= e^{8 y l n (x)} \frac{d}{d y} (8 y ln (x))

\frac{d}{d y} (8 y ln (x)) = 8 ln (x)

= e^{8 y l n (x)} \cdot 8 ln (x)

e^{8 y l n (x)} = x^{8 y}

= 8 x^{8 y} ln (x)