What is the derivative of (sqrt(x)^{5x}) ?

The derivative of (sqrt(x)^{5x}) is 5x^{(5x)/2}(1/2 ln(x)+1/2)

What is the first derivative of (sqrt(x)^{5x}) ?

The first derivative of (sqrt(x)^{5x}) is 5x^{(5x)/2}(1/2 ln(x)+1/2)

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\frac{d}{d x} ((x)^{5 x})

5 x^{\frac{5 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

Solution steps

\frac{d}{d x} ((x)^{5 x})

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d x} (5 x ln (x)) = 5 (ln (x) + \frac{1}{2})

= e^{5 x l n (x)} \cdot 5 (ln (x) + \frac{1}{2})

Simplify

e^{5 x l n (x)} \cdot 5 (ln (x) + \frac{1}{2}) : 5 x^{\frac{5 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

= 5 x^{\frac{5 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

\int \frac{x ^{3} + x + 4}{x ^{2}} d x

\frac{d}{d y} (arctan (\frac{x}{y}))

\frac{d}{d x} (arctan (x^{4}))

f (t) = \frac{t ^{2} + 1}{t ^{2} - 1}

What is the derivative of (sqrt(x)^{5x}) ?
The derivative of (sqrt(x)^{5x}) is 5x^{(5x)/2}(1/2 ln(x)+1/2)
What is the first derivative of (sqrt(x)^{5x}) ?
The first derivative of (sqrt(x)^{5x}) is 5x^{(5x)/2}(1/2 ln(x)+1/2)