Question

Is d2y/dx2=dy/dx*dy/dx

Answer 1

 d2y/dx2=dy/dx*dy/dx
this is wrong

 d2y/dx2 =  d/dx(dy/dx)
this is correct

Answer 2

No.

$\boxed{\mathsf{\frac{d^{2}y}{dx^{2}}\neq \frac{dy}{dx}*\frac{dy}{dx}}}$ not necessarily!

We can show that by taking two examples of two types.

• Example 1.

Let, $\mathsf{y=x^{3}+\frac{1}{2}x^{2}+x}$

Differentiating both sides with respect to x, we get

   $\mathsf{\frac{dy}{dx}=3x^{2}+x+1}$

Again, Differentiating both sides with respect to x, we get

   $\mathsf{\frac{d^{2}y}{dx^{2}}=6x+1}$

$\mathsf{\frac{dy}{dx}*\frac{dy}{dx}=(3x^{2}+2x+1)^{2}}$

$=\mathsf{9x^{4}+4x^{2}+1+12x^{3}+6x^{2}+4x}$

$=\mathsf{9x^{4}+12x^{3}+10x^{2}+4x+1}$

Hence, $\boxed{\mathsf{\frac{d^{2}y}{dx^{2}}\neq \frac{dy}{dx}*\frac{dy}{dx}}}$

• Example 2.

Now, we take any constant function.

Let, y = a

Differentiating with respect to x, we get

   $\mathsf{\frac{dy}{dx}=0}$

So, $\mathsf{\frac{dy}{dx}*\frac{dy}{dx}=0}$

Again, Differentiating with respect to x, we get

   $\mathsf{\frac{d^{2}x}{dx^{2}}=0}$

In this case, $\boxed{\mathsf{\frac{d^{2}y}{dx^{2}}=\frac{dy}{dx}*\frac{dy}{dx}}}$

Note : In general, we cannot write $\mathsf{\frac{d^{2}y}{dx^{2}}=\frac{dy}{dx}*\frac{dy}{dx}}$ because $\mathsf{\frac{d^{2}y}{dx^{2}}}$ is double derivative of a function y=f(x), where $\mathsf{(\frac{dy}{dx})^{2}}$ is the product of two first order derivatives of y=f(x).