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Popular >

derivative of y=(sqrt(x))^{3x}

Solution

derivative of

y = (x)^{3 x}

Solution

3 (x)^{3 x} (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})

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

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

Popular Examples

derivative of xe^{-x}+e^{3x}

\frac{d}{d x} (x e^{- x} + e^{3 x})

integral of 1-cos(2θ)

\int 1 - cos (2 θ) d θ

x y^{^{'}} - y = x^{3} y^{4}

(\partial)/(\partial x)(3cos(x)-sin(xy))

\frac{\partial}{\partial x} (3 cos (x) - sin (x y))

integral of (sin(2x))/((1-cos^2(x)))

\int \frac{sin ( 2 x )}{( 1 - cos ^{2} ( x ) )} d x

Frequently Asked Questions (FAQ)

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