If y is a function of z and z = ax, then prove that d^2y/dx^2=a^2(d^2y/dx^2)

Let y=f(x) , dy/dx = f'(x) and d²y/dx²= f''(x)

Z=ax

We know dy/dz can also be written as

(Dy/dx)÷(dz/dx)

Now if we differentiate the numerator and denominator

We get dy/dz=(dy/dx)÷a = f'(x)/a

Differentiating the the above equation again

We get ,

d²y/dz² = [ d(f'(x))/dz * a - f'(x)*d(a)/dz ] ÷ a² (APPLYING QUOTIENT RULE)

a²*d²y/dz² = d(f'(x))/dx *a *dx/dz - f'(x) *0 (APPLYING CHAIN RULE)

a²*d²x/dz² = f''(x) *a* (dx / d(ax)). (Z=ax)

a²*d²x/dz² = f''(x) *a*(dx/a * dx) (a is taken outside Being a constant , contsant rule)

Cutting a and dx from numerator and denominator (i assume that a is contsant and is not equal to zero. This information is a must. Otherwise the whole question is itself invalid)

We get our final result as

a²*d²x/dz² = f''(x)

= a² * d²x/dz² = d²y/dx²

Hence we have proved the given question!