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

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

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

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

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derivative of

(x)^{5 x}

5 (x)^{5 x} (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})

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

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

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