Question

If $y=a\sin (5x+c)$, then

• A. $\frac{dy}{dx}=5y$
• B. $\frac{dy}{dx}=-5y$
• C. $\frac{d^{2}y}{d{x}^2}=-25y$
• D. $\frac{d^{2}y}{d{x}^2}=25y$

Answer 1

The correct option is C

$\frac{d^{2}y}{d{x}^2}=-25y$

Explanation for the correct option.

Step 1. Find the value of $\frac{dy}{dx}$.

Differentiate the function $y=a\sin (5x+c)$ with respect to $x$.

$\begin{array}{rcl}\frac{dy}{dx}& =& \frac{d}{dx}\left(a\sin \left(5x+c\right)\right)\\ & =& a\cos \left(5x+c\right)\times 5\left[\because \frac{d}{\mathrm{dx}}\left(\mathrm{sinx}\right)=\mathrm{cosx}\right]\\ & =& 5a\cos \left(5x+c\right)\end{array}$

Step 2. Find the value of $\frac{d^{2}y}{d{x}^2}$.

Differentiate the function $\frac{dy}{dx}=5a\cos \left(5x+c\right)$ with respect to $x$.

$\begin{array}{rcl}\frac{d^{2}y}{d{x}^2}& =& \frac{d}{dx}\left(5a\cos \left(5x+c\right)\right)\\ & =& 5a\left(-\sin \left(5x+c\right)\right)\times 5\left[\because \frac{d}{\mathrm{dx}}\left(\mathrm{cosx}\right)=-\mathrm{sinx}\right]\\ & =& -25a\sin \left(5x+c\right)\\ & =& -25y\left[y=a\sin \left(5x+c\right)\right]\end{array}$

Hence, the correct option is C.