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

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

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

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

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derivative of

y = (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})

\int 9 (3 x - 2)^{2} d x

\frac{d ^{3}}{d x ^{3}} (\frac{5}{x ^{2}})

f (x) = (x^{2} - 3 x + 8)^{3}

x \to - \infty lim (x^{2} + 1)

y^{^{''}} - 2 y^{^{'}} - 3 y = - 5 x e^{4 x}

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