Find the value of $\dfrac{dy}{dx}$ at $\theta=\dfrac{\pi}{4}$, if $x=a\,e^{\theta}(\sin\theta-\cos\theta)$ and $y=a\,e^{\theta}(\sin\theta+\cos\theta)$. (2014)

$\dfrac{dy}{dx}=\cot\theta$; at $\theta=\dfrac{\pi}{4}$ it is $1$.

CBSE Class 12 · 2014 · DifferentiationLong Answer

Find the value of

\frac{d y}{d x}

θ = \frac{π}{4}

, if

x = a e^{θ} (sin θ - cos θ)

and

y = a e^{θ} (sin θ + cos θ)

. (2014)

Answer: $\frac{d y}{d x} = cot θ$ ; at $θ = \frac{π}{4}$ it is $1$ .

Use parametric differentiation:

\frac{d y}{d x} = \frac{d y / d θ}{d x / d θ}

.
Differentiate

y

(product rule):

\frac{d y}{d θ} = a e^{θ} (sin θ + cos θ) + a e^{θ} (cos θ - sin θ) = 2 a e^{θ} cos θ .

Differentiate

x

\frac{d x}{d θ} = a e^{θ} (sin θ - cos θ) + a e^{θ} (cos θ + sin θ) = 2 a e^{θ} sin θ .

Therefore

\frac{d y}{d x} = \frac{2 a e ^{θ} cos θ}{2 a e ^{θ} sin θ} = cot θ .

θ = \frac{π}{4}

cot \frac{π}{4} = 1

\frac{d y}{d x}_{θ = π /4} = 1

CBSE Class 12 Board · 2014 · Subjective