What is the derivative of f(x)=(3x)^{2x} ?

The derivative of f(x)=(3x)^{2x} is 2ln(3)*9^xx^{2x}+2x^{2x}(ln(x)+1)*9^x

What is the first derivative of f(x)=(3x)^{2x} ?

The first derivative of f(x)=(3x)^{2x} is 2ln(3)*9^xx^{2x}+2x^{2x}(ln(x)+1)*9^x

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derivative of

f (x) = (3 x)^{2 x}

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

Solution steps

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

Simplify

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

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

Apply the Product Rule:

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

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

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

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

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

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

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What is the derivative of f(x)=(3x)^{2x} ?
The derivative of f(x)=(3x)^{2x} is 2ln(3)*9^xx^{2x}+2x^{2x}(ln(x)+1)*9^x
What is the first derivative of f(x)=(3x)^{2x} ?
The first derivative of f(x)=(3x)^{2x} is 2ln(3)*9^xx^{2x}+2x^{2x}(ln(x)+1)*9^x