What is the derivative of y=(3x)^{5x} ?

The derivative of y=(3x)^{5x} is 5ln(3)*243^xx^{5x}+5x^{5x}(ln(x)+1)*243^x

What is the first derivative of y=(3x)^{5x} ?

The first derivative of y=(3x)^{5x} is 5ln(3)*243^xx^{5x}+5x^{5x}(ln(x)+1)*243^x

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derivative of

y = (3 x)^{5 x}

5 ln (3) \cdot 24 3^{x} x^{5 x} + 5 x^{5 x} (ln (x) + 1) \cdot 24 3^{x}

Solution steps

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

Simplify

(3 x)^{5 x} : 24 3^{x} x^{5 x}

= \frac{d}{d x} (24 3^{x} x^{5 x})

Apply the Product Rule:

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

f = 24 3^{x}, g = x^{5 x}

= \frac{d}{d x} (24 3^{x}) x^{5 x} + \frac{d}{d x} (x^{5 x}) 24 3^{x}

\frac{d}{d x} (24 3^{x}) = 5 ln (3) \cdot 24 3^{x}

\frac{d}{d x} (x^{5 x}) = 5 x^{5 x} (ln (x) + 1)

= 5 ln (3) \cdot 24 3^{x} x^{5 x} + 5 x^{5 x} (ln (x) + 1) \cdot 24 3^{x}

What is the derivative of y=(3x)^{5x} ?
The derivative of y=(3x)^{5x} is 5ln(3)*243^xx^{5x}+5x^{5x}(ln(x)+1)*243^x
What is the first derivative of y=(3x)^{5x} ?
The first derivative of y=(3x)^{5x} is 5ln(3)*243^xx^{5x}+5x^{5x}(ln(x)+1)*243^x