What is (\partial)/(\partial x)(y^{5x}) ?

The answer to (\partial)/(\partial x)(y^{5x}) is 5y^{5x}ln(y)

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(\partial)/(\partial x)(y^{5x})

\frac{\partial}{\partial x} (y^{5 x})

5 y^{5 x} ln (y)

Solution steps

\frac{\partial}{\partial x} (y^{5 x})

Treat

y

as a constant

Apply exponent rule:

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

y^{5 x} = e^{5 x l n (y)}

= \frac{\partial}{\partial x} (e^{5 x l n (y)})

Apply the chain rule:

e^{5 x l n (y)} \frac{\partial}{\partial x} (5 x ln (y))

= e^{5 x l n (y)} \frac{\partial}{\partial x} (5 x ln (y))

\frac{\partial}{\partial x} (5 x ln (y)) = 5 ln (y)

= e^{5 x l n (y)} \cdot 5 ln (y)

e^{5 x l n (y)} = y^{5 x}

= 5 y^{5 x} ln (y)

\frac{d}{d x} (4 - e^{x})

\frac{d}{d x} (\frac{( 4 x ^{2} + 2 x + 2 )}{x})

y^{^{''}} + 6 y^{^{'}} = 162 e^{3 t}, y (0) = 6, y^{^{'}} (0) = 7

t \to \infty lim (\frac{0.2 t}{t ^{2} + 1})

\frac{d}{d x} (2 (- 2 e^{- x^{2}} x^{2} + e^{- x^{2}}))

What is (\partial)/(\partial x)(y^{5x}) ?
The answer to (\partial)/(\partial x)(y^{5x}) is 5y^{5x}ln(y)