When differentiating the equation with respect to

MATHEMATICS · Single correct · A Level · Mathematics · Differentiation · Differentiation

Problem

When differentiating the equation

x^{2}+y^{2}=25

with respect to

x

, which of the following best explains why the derivative of the term

y^{2}

is written as

2y\,\dfrac{dy}{dx}

rather than simply

2y

? (a) Because implicit differentiation rules that every term containing

y

must be multiplied by

\dfrac{dy}{dx}

after differentiating. (b) Because

y

is treated as a function of

x

, so the chain rule introduces the factor

\dfrac{dy}{dx}

. (c) Because treating

y

as a constant with respect to

x

gives the derivative

2y

, so no additional factor is necessary. (d) Because the power rule is applied twice—once to obtain

2y

and again to differentiate

y

itself—resulting in the extra factor

\dfrac{dy}{dx}