What is the derivative of 3^{2x^2}*sqrt(x) ?

The derivative of 3^{2x^2}*sqrt(x) is 4ln(3)*9^{x^2}xsqrt(x)+(9^{x^2})/(2sqrt(x))

What is the first derivative of 3^{2x^2}*sqrt(x) ?

The first derivative of 3^{2x^2}*sqrt(x) is 4ln(3)*9^{x^2}xsqrt(x)+(9^{x^2})/(2sqrt(x))

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

4 ln (3) \cdot 9^{x^{2}} x x + \frac{9 ^{x^{2}}}{2 x}

Solution steps

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

Simplify

3^{2 x^{2}} x : 9^{x^{2}} x

= \frac{d}{d x} (9^{x^{2}} x)

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = 9^{x^{2}}, g = x

= \frac{d}{d x} (9^{x^{2}}) x + \frac{d}{d x} (x) 9^{x^{2}}

\frac{d}{d x} (9^{x^{2}}) = 4 ln (3) \cdot 9^{x^{2}} x

\frac{d}{d x} (x) = \frac{1}{2 x}

= 4 ln (3) \cdot 9^{x^{2}} x x + \frac{1}{2 x} \cdot 9^{x^{2}}

\frac{1}{2 x} \cdot 9^{x^{2}} = \frac{9 ^{x^{2}}}{2 x}

= 4 ln (3) \cdot 9^{x^{2}} x x + \frac{9 ^{x^{2}}}{2 x}

\frac{d}{d x} (\frac{10}{x ^{2} + 1})

x \to - 4 lim (\frac{1}{2} x - 1)

\frac{7 e ^{x} x ^{\frac{5}{2}}}{2} + e^{x} x^{\frac{7}{2}}

\int 0.4 \cdot e^{- 2000 x} d x

\frac{\partial}{\partial x} (\frac{x ^{2} + 2}{x ^{2} - 2})

What is the derivative of 3^{2x^2}*sqrt(x) ?
The derivative of 3^{2x^2}*sqrt(x) is 4ln(3)*9^{x^2}xsqrt(x)+(9^{x^2})/(2sqrt(x))
What is the first derivative of 3^{2x^2}*sqrt(x) ?
The first derivative of 3^{2x^2}*sqrt(x) is 4ln(3)*9^{x^2}xsqrt(x)+(9^{x^2})/(2sqrt(x))