What is the derivative of (sqrt(x)^{3x}) ?

The derivative of (sqrt(x)^{3x}) is 3x^{(3x)/2}(1/2 ln(x)+1/2)

What is the first derivative of (sqrt(x)^{3x}) ?

The first derivative of (sqrt(x)^{3x}) is 3x^{(3x)/2}(1/2 ln(x)+1/2)

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\frac{d}{d x} ((x)^{3 x})

3 x^{\frac{3 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

Solution steps

\frac{d}{d x} ((x)^{3 x})

Apply exponent rule:

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

(x)^{3 x} = e^{3 x l n (x)}

= \frac{d}{d x} (e^{3 x l n (x)})

Apply the chain rule:

e^{3 x l n (x)} \frac{d}{d x} (3 x ln (x))

= e^{3 x l n (x)} \frac{d}{d x} (3 x ln (x))

\frac{d}{d x} (3 x ln (x)) = 3 (ln (x) + \frac{1}{2})

= e^{3 x l n (x)} \cdot 3 (ln (x) + \frac{1}{2})

Simplify

e^{3 x l n (x)} \cdot 3 (ln (x) + \frac{1}{2}) : 3 x^{\frac{3 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

= 3 x^{\frac{3 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

\frac{d y}{d x} = \frac{- y - 6 x ^{2}}{x + 3}

x \to 3 - lim (h (x))

f (x) = 6 x

\int_{0}^{1} \frac{x + 1}{x ^{2} + 2 x - 3} d x

\frac{d}{d x} (a x^{2} (x - 4) + b x^{4} ln (\frac{x}{4}))

What is the derivative of (sqrt(x)^{3x}) ?
The derivative of (sqrt(x)^{3x}) is 3x^{(3x)/2}(1/2 ln(x)+1/2)
What is the first derivative of (sqrt(x)^{3x}) ?
The first derivative of (sqrt(x)^{3x}) is 3x^{(3x)/2}(1/2 ln(x)+1/2)