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Popular >

derivative of (sqrt(x))^{5x}

Solution

derivative of

(x)^{5 x}

Solution

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})

Popular Examples

((x^2+1)dy)/(dx)+3x(y-1)=0

\frac{( x ^{2} + 1 ) d y}{d x} + 3 x (y - 1) = 0

derivative of arcsin((x^2+y^2/2))

\frac{d}{d x} (arcsin (\frac{x ^{2} + y ^{2}}{2}))

tangent of y=(6x)/(x^2+1),(1,3)tangent of

y = \frac{6 x}{x ^{2} + 1}, (1, 3)

integral of 1/(2*sqrt(x))

\int \frac{1}{2 \cdot x} d x

taylor 1/(x+3)taylor

\frac{1}{x + 3}

Frequently Asked Questions (FAQ)

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)