Question

If y=sinxcosx, then at x=pi/3, dy/dx=

Please include the steps to solve this problem.

Answer 1

The answer is - $\frac{1}{2}$

Given: y = sin(x) cos(x)

Applying the product rule of differentiation, we get,

$\frac{dy}{dx}$= sin(x) . $\frac{dy}{dx}$ cos(x) + cos(x)$\frac{dy}{dx}$ sin(x)

                                   = sin(x) × -sin(x) + cos(x)cos(x)
 
                                   = - sin²x + cos²x
       
                                   = - sin²x + (1- sin²x)
                       
                                  = 1 - 2sin²x = cos 2x

Now, we need to find, $\frac{dy}{dx} at x = \frac{π}{3}$
 
$\frac{dy}{dx}$ = cos (2×60) = cos (120)

⇒$\frac{dy}{dx} = - \frac{1}{2}$

Answer 2

The value of $dy/dx$ at $x=\pi/3$ is (-1/2).

Step-by-step explanation:

Given information:

The expression $y=sinx.cosx$

And $x= \pi/3$

Now to find derivative $dy/dx$ of the given expression

Apply product rule for differentiation :

As,

$y=sinx.cosx\\\\\frac{dy}{dx} = (sinx .\frac{dy}{dx}cosx)+ (cosx.\frac{dy}{dx}sinx)\\\\\frac{dy}{dx}=(sinx.(-sinx))+(cosx.cosx)\\\\\frac{dy}{dx}=-sin^2x+cos^2x\\\\\frac{dy}{dx} =-sin^2x+(1-sin^2x)\\\\\frac{dy}{dx}=1-2sin^2x\\\\\frac{dy}{dx}=cos2x$

Now , at $x= \pi/3$

$dy/dx=cos2x\\dy/dx=cos(2 \times 60)\\dy/dx=cos120\\dy/dx=(-1/2)$

Hence , the value of $dy/dx$ at $x=\pi/3$ is (-1/2).

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