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.