To determine

To find: The formula for d3ydx3.

Answer

The formula for d3ydx3 is d3ydx3=dydu(d3udx3)+3dudx(d2udx2)(d2ydu2)+(dudx)3(d3ydu3).

Explanation

Result used: Chain Rule

If y=f(u) and u=g(x) are both differentiable function, then dydx=dydududx.     (1)

Derivative Rule:

Product Rule: ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)]

Calculation:

Let y=f(u) and u=g(x).

The derivative d3ydx3 is computed as follows,

d3ydx3=ddx(d2ydx2) (1)

Obtain the derivative d2ydx2.

d2ydx2=ddx(dydx)=ddx(dydududx)        (Chain Rule(1))

Apply the product rule (1),

d2ydx2=dyduddx(dudx)+dudxddx(dydu)=dydud2udx2+dudx(d2ydu2dudx)=dydud2udx2+d2ydu2dudxdudx=dydud2udx2+d2ydu2(dudx)2

Substitute dydud2udx2+d2ydu2(dudx)2 for d2ydx2 in equation (1),

d3ydx3=ddx(dydud2udx2+d2ydu2(dudx)2)=ddx(dydud2udx2)+ddx(d2ydu2(dudx)2)

Apply the product rule (1),

d3ydx3=[dyduddx(d2udx2)+(d2udx2)ddx(dydu)]+[d2ydu2ddx(dudx)2+(dudx)2ddx(d2ydu2)]=[dydu(d3udx3)+(d2udx2)(d2ydu2dudx)]+[d2ydu2(2(dudx)2−1ddx(dudx))+(dudx)2(d3ydu3dudx)]=[dydu(d3udx3)+(d2udx2)(d2ydu2dudx)]+[d2ydu2(2(dudx)(d2udx2))+(dudx)3(d3ydu3)]=dydu(d3udx3)+3dudx(d2udx2)(d2ydu2)+(dudx)3(d3ydu3)

Therefore, the derivative d3ydx3=dydu(d3udx3)+3dudx(d2udx2)(d2ydu2)+(dudx)3(d3ydu3).