The frequency of vibrations of vibrating violin string is .

Where is the length of the string, is tension of the string and is linear density of the string.

(a)

(i)

Rate of change of frequency  with respect to length (when and are constants):

Consider .

Differentiate on each side with respect to .

.

(ii)

Rate of change of frequency  with respect to tension (when and are constants):

Consider .

Differentiate on each side with respect to .

.

(iii)

Rate of change of frequency with respect to the linear density (when and are constants):

Consider .

Differentiate on each side with respect to .

.

(b)

(i)

When the effective length of string is decreased :

Since , it is concluded that is a decreasing function with respect to length.

Hence if is decreasing , then will increases.

Thus, the pitch is increasing.

(ii)

When the tension of the string is increases :

Since , it is concluded that is a increasing function with respect to tension .

If is increasing, then will increases.

Thus, the pitch is increasing.

(iii)

When the linear density is  increased:

Since , it is concluded that is a decreasing function with respect to linear density .

If is increasing , then the will also decreases.

Thus, the pitch is decreasing.

(a) (i) , (ii) and (iii) .

(b)

(i) The pitch is increasing.

(ii) The pitch is increasing.

(iii) The pitch is decreasing.