Which of the following best explains why the derivative of equals rather than simply ?

MATHEMATICS · Single correct · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Which of the following best explains why the derivative of

2^{x}

equals

2^{x}\ln 2

rather than simply

2^{x}

? (a) Because

2^{x}

grows more slowly than

e^{x}

, its derivative must be

\dfrac{2^{x}}{\ln 2}

to scale the rate of change. (b) Because the rate of change of

2^{x}

is proportional to its value, with proportionality constant

\ln 2

that depends on the base 2. (c) Because differentiating

2^{x}

is treated like the power rule, so the exponent

x

comes down, giving

x2^{x-1}

, which is written as

2^{x}\ln 2

when

x=1

. (d) Because writing

2^{x}=(e^{x})^{\ln 2}

and applying the power rule yields

(\ln 2)\,2^{x-1}

, so the factor

\ln 2

appears in the derivative.