Question

Find the differential of the function \(m = {p^5}{q^3}\).

Short answer

The differential of the function \(m = {p^5}{q^3}\) is

\(dm = 5{p^4}{q^3}dp + 3{p^5}{q^2}dq\).

Step-by-step solution

Finding Partial Differentiation of \(p\).

Given:\(m = {p^5}{q^3} = f\left( {p,q} \right)\)

Differentiating both sides

\(dm = {f_p}\left( {p,q} \right)dp + {f_q}\left( {p,q} \right)dq\)…… (1)

Now, \(f\left( {p,q} \right) = {p^5}{q^3}\)

Partial differentiating with respect to \(p\).

\({f_p} = 5{p^4}{q^3}\)

Step2: Partial Differentiating with respect to \(q\) and putting values in equation (1).

\(f\left( {p,q} \right) = {p^5}{q^3}\)

Partial differentiating with respect to \(q\).

\({f_q} = 3{p^5}{q^2}\)

Substituting the values of\({f_p}\)and \({f_q}\) in (1).

\(dm = 5{p^4}{q^3}dp + 3{p^5}{q^2}dq\)

Hence after differentiating the function \(m = {p^5}{q^3}\) is \(dm = 5{p^4}{q^3}dp + 3{p^5}{q^2}dq\).