What is the derivative of 5^{3x^3} ?

The derivative of 5^{3x^3} is 9ln(5)*125^{x^3}x^2

What is the first derivative of 5^{3x^3} ?

The first derivative of 5^{3x^3} is 9ln(5)*125^{x^3}x^2

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

9 ln (5) \cdot 12 5^{x^{3}} x^{2}

Solution steps

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

Simplify

5^{3 x^{3}} : 12 5^{x^{3}}

= \frac{d}{d x} (12 5^{x^{3}})

Apply exponent rule:

a^{b} = e^{b l n (a)}

12 5^{x^{3}} = e^{x^{3} l n (125)}

= \frac{d}{d x} (e^{x^{3} l n (125)})

Apply the chain rule:

e^{x^{3} l n (125)} \frac{d}{d x} (x^{3} ln (125))

= e^{x^{3} l n (125)} \frac{d}{d x} (x^{3} ln (125))

\frac{d}{d x} (x^{3} ln (125)) = 9 ln (5) x^{2}

= e^{x^{3} l n (125)} \cdot 9 ln (5) x^{2}

e^{x^{3} l n (125)} = 12 5^{x^{3}}

= 9 ln (5) \cdot 12 5^{x^{3}} x^{2}

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What is the derivative of 5^{3x^3} ?
The derivative of 5^{3x^3} is 9ln(5)*125^{x^3}x^2
What is the first derivative of 5^{3x^3} ?
The first derivative of 5^{3x^3} is 9ln(5)*125^{x^3}x^2