KCET 2025 Math Paper

To find the derivative of sin‌x with respect to log‌x, we need to use the concept of implicit differentiation and the chain rule.
Let f(x)=sin‌x.
We want to find ‌

d(sin‌x)

d(log‌x)

.
Using the chain rule, we have:
‌

d(sin‌x)

d(log‌x)

=‌

d(sin‌x)

dx

⋅‌

dx

d(log‌x)

First, we calculate each part separately:
‌

d(sin‌x)

dx

=cos‌x
To find ‌

dx

d(log‌x)

, consider that ‌

d(log‌x)

dx

=‌

1

x

, hence:
‌

dx

d(log‌x)

=x
Putting it all together, we get:
‌

d(sin‌x)

d(log‌x)

=cos‌x⋅x=x‌cos‌x
Therefore, the derivative of sin‌x with respect to log‌x is x‌cos‌x.