To find the value of
‌, we start with the given relationships:
y=f(x),‌‌p=‌,‌‌q=‌First, we know from the given definitions:
p=‌ Taking the reciprocal of this derivative gives:
‌=‌Next, we need to differentiate
‌ with respect to
y to find the second derivative of
x with respect to
y.
‌=‌(‌)=‌(‌) Using the chain rule, we can write:
‌(‌)=‌(‌)⋅‌We already have
‌=‌, so we need to find
‌(‌). Using the chain rule for differentiation:
‌(‌)=−‌⋅‌ We need the derivative of
p with respect to
x, which is given by the second derivative of
y with respect to
x :
‌=‌=qSubstituting this into our equation:
‌(‌)=−‌ Now plugging this back into the earlier expression for
‌ :
‌=−‌⋅‌=−‌Therefore, the correct option is:
Option C:
−‌