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

The derivative of 3a^{5x} is 15a^{5x}ln(a)

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

The first derivative of 3a^{5x} is 15a^{5x}ln(a)

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

15 a^{5 x} ln (a)

Solution steps

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply exponent rule:

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

a^{5 x} = e^{5 x l n (a)}

= 3 \frac{d}{d x} (e^{5 x l n (a)})

Apply the chain rule:

e^{5 x l n (a)} \frac{d}{d x} (5 x ln (a))

= e^{5 x l n (a)} \frac{d}{d x} (5 x ln (a))

\frac{d}{d x} (5 x ln (a)) = 5 ln (a)

= 3 e^{5 x l n (a)} \cdot 5 ln (a)

Simplify

3 e^{5 x l n (a)} \cdot 5 ln (a) : 15 a^{5 x} ln (a)

= 15 a^{5 x} ln (a)

What is the derivative of 3a^{5x} ?
The derivative of 3a^{5x} is 15a^{5x}ln(a)
What is the first derivative of 3a^{5x} ?
The first derivative of 3a^{5x} is 15a^{5x}ln(a)