 Hi and welcome to the session. I am Vipika here. Let's discuss a question if u, v and w are functions of x then show that d by dx of u, v, w is equal to du by dx into v into w plus u into dv by dx into w plus u into v into dw by dx in two ways First by repeated application of product rule taken by logarithmic differentiation. So let's start the solution. First we will do with repeated application of product rule. That y is equal to u dot v w difference here both sides with respect to x d y by dx is equal to d by dx of u v w let us take u v is equal to z. So therefore d y by dx is equal to d by dx of z m w z into w now we get d y by dx is equal to z into d w by dx plus w into d z by dx now put value of z here that is u v is equal to z we get d y by dx is equal to u v into d w u v into d w by dx plus w into d by dx of z that is u v again here we will apply the product rule so we have d y by dx is equal to u v d w by dx plus w into u into derivative of v that is d v by dx plus v into derivative of u that is d u by dx so we have d y by dx is equal to this is u v into d w by dx plus w u into d v by dx plus v w into d u by dx hence d by dx of u v w is equal to d u by dx into v into w plus u into d v by dx into w plus u into v into d w by dx now let us differentiate the same function with the second way that is by repeated by logarithmic differentiation so again why we have u v w so taking logarithm on both sides both sides we have log y is equal to log u plus log v plus log w now differentiate both sides with respect to x we get 1 by y into d y by dx is equal to 1 by u into derivative of u with respect to x is d u by dx plus 1 by v into derivative of v with respect to x that is d v by dx plus 1 over w into d w by dx this implies d y by dx is equal to y into 1 by u into d u by dx plus 1 by v into d v by dx plus 1 by w into d w by dx so now substitute the value of y here we get d y by dx is equal to u v w into 1 by u into d u by dx plus 1 by v into d v by dx plus 1 by w into d w by dx therefore d by dx of u v w is equal to d u by dx into v into w plus u into d v by dx into w plus u into v into d w by dx hence we have proved the above function by two ways first by repeated application of product rule and second by logarithmic differentiation I hope the question is clear to you buy and have a nice day.