What is the derivative of (5x)^{9x} ?

The derivative of (5x)^{9x} is 9ln(5)*1953125^xx^{9x}+9x^{9x}(ln(x)+1)*1953125^x

What is the first derivative of (5x)^{9x} ?

The first derivative of (5x)^{9x} is 9ln(5)*1953125^xx^{9x}+9x^{9x}(ln(x)+1)*1953125^x

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derivative of

(5 x)^{9 x}

9 ln (5) \cdot 195312 5^{x} x^{9 x} + 9 x^{9 x} (ln (x) + 1) \cdot 195312 5^{x}

Solution steps

\frac{d}{d x} ((5 x)^{9 x})

Simplify

(5 x)^{9 x} : 195312 5^{x} x^{9 x}

= \frac{d}{d x} (195312 5^{x} x^{9 x})

Apply the Product Rule:

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

f = 195312 5^{x}, g = x^{9 x}

= \frac{d}{d x} (195312 5^{x}) x^{9 x} + \frac{d}{d x} (x^{9 x}) 195312 5^{x}

\frac{d}{d x} (195312 5^{x}) = 9 ln (5) \cdot 195312 5^{x}

\frac{d}{d x} (x^{9 x}) = 9 x^{9 x} (ln (x) + 1)

= 9 ln (5) \cdot 195312 5^{x} x^{9 x} + 9 x^{9 x} (ln (x) + 1) \cdot 195312 5^{x}

What is the derivative of (5x)^{9x} ?
The derivative of (5x)^{9x} is 9ln(5)*1953125^xx^{9x}+9x^{9x}(ln(x)+1)*1953125^x
What is the first derivative of (5x)^{9x} ?
The first derivative of (5x)^{9x} is 9ln(5)*1953125^xx^{9x}+9x^{9x}(ln(x)+1)*1953125^x