What is the derivative of sqrt(x)^{9x} ?

The derivative of sqrt(x)^{9x} is 9x^{(9x)/2}(1/2 ln(x)+1/2)

What is the first derivative of sqrt(x)^{9x} ?

The first derivative of sqrt(x)^{9x} is 9x^{(9x)/2}(1/2 ln(x)+1/2)

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\frac{d}{d x} (x^{9 x})

9 x^{\frac{9 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

Solution steps

\frac{d}{d x} ((x)^{9 x})

Apply exponent rule:

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

(x)^{9 x} = e^{9 x l n (x)}

= \frac{d}{d x} (e^{9 x l n (x)})

Apply the chain rule:

e^{9 x l n (x)} \frac{d}{d x} (9 x ln (x))

= e^{9 x l n (x)} \frac{d}{d x} (9 x ln (x))

\frac{d}{d x} (9 x ln (x)) = 9 (ln (x) + \frac{1}{2})

= e^{9 x l n (x)} \cdot 9 (ln (x) + \frac{1}{2})

Simplify

e^{9 x l n (x)} \cdot 9 (ln (x) + \frac{1}{2}) : 9 x^{\frac{9 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

= 9 x^{\frac{9 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

\int_{0}^{1} (x^{2} - 8) e^{- x} d x

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

x^{2} y y^{^{'}} = (y^{2} - 1)^{\frac{3}{2}}

\frac{d}{d x} (x ln (\frac{1}{x}))

y = 2

What is the derivative of sqrt(x)^{9x} ?
The derivative of sqrt(x)^{9x} is 9x^{(9x)/2}(1/2 ln(x)+1/2)
What is the first derivative of sqrt(x)^{9x} ?
The first derivative of sqrt(x)^{9x} is 9x^{(9x)/2}(1/2 ln(x)+1/2)