"What is the derivative of 3x ?" solved and explained

Ashley Bell

Answered question

2021-12-14

What is the derivative of

3^{x}

abonirali59

Beginner2021-12-15Added 35 answers

Begin by letting

y = 3^{x}

now take the ln of both sides.

\ln y = {\ln 3}^{x} \Rightarrow \ln y = x \ln 3

differentiate implicitly with respect to x

\Rightarrow \frac{1}{y} \frac{d y}{d x} = \ln 3

Now

y = 3^{x} \Rightarrow \frac{d y}{d x} = 3^{x} \ln 3

This result can be generalised as follows

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

psor32

Beginner2021-12-16Added 33 answers

Explanation:

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

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

(since

a^{b} = e^{b \ln (a)}

)

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

= \ln (3) e^{x \ln (3)}

= \ln (3) 3^{x}

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