Question

How do you find the second derivative of $y=A\sin \left( Bx \right)$?

Answer 1

In the given question, we need to find out the second derivative. A second order derivative is a derivative of a function and it is drawn from the first-order derivative. So we first need to find out the derivative of a function and then draw out the derivative of the first derivative. A first order derivative can be written as $f'x$ or $\dfrac{dy}{dx}$ whereas the second order derivative can be written as $f''x$ or $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$. Direct apply the derivative and apply the necessary rules of differentiation.

Formula used:
$\dfrac{d}{dx}\sin \left( Kx \right)=K\cos \left( Kx \right)$
$\dfrac{d}{dx}\cos \left( Kx \right)=-K\sin \left( Kx \right)$

Complete step by step answer:
We have the following function:
$y=A\sin \left( Bx \right)$
Differentiating the following function, we get
$\dfrac{dy}{dx}=A\left( B\cos \left( Bx \right) \right)$
$\Rightarrow\dfrac{dy}{dx}=AB\cos \left( Bx \right)$
Differentiating the first order derivative again in order to determine the second order derivative, we get
$\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=AB\left( -B\sin \left( Bx \right) \right)$
$\Rightarrow\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=-A{{B}^{2}}\sin \left( Bx \right)$
$\Rightarrow -A{{B}^{2}}\sin \left( Bx \right)$
$\therefore -A{{B}^{2}}\sin \left( Bx \right)$ is the second derivative of a given function $y=A\sin \left( Bx \right)$.

Note: Differentiating helps us to determine the rates of change. There are a number of simple rules which can be used to differentiate the functions. A second order derivative can be used to determine the concavity and the inflexion points. Second order derivatives tell us about the function that can either be concave up or concave down. When the second order derivative of a function is positive, the function will be concave up and when the second order derivative of a function is negative, the function will be concave down. If the second order derivative of a function tends to 0, then the function can either be concave up or concave down or keep shifting.