Find the value of dy - dx at -4-if x - ae -sin-cos- andy - ae -sin-cos-

x=ae^θ(sin θ cos θ) dx/dθ=ae^θ(sin θ cos θ)+ae^θ(sin θ+cos θ) =2ae^θsinθ y = ae^θ (sin θ + cos θ) dy/dθ=ae^θ(sin θ+cos θ)+ae^θ(cos θ sin θ) =2ae^θcosθ dy ...

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Byju's Answer

Standard XII

Mathematics

Integration by Parts

Find the valu...

Question

Find the value of $\frac{d y}{d x}$ at $θ = \frac{π}{4}$ ,if $x = a e^{θ} (s i n θ - c o s θ)$ and $y = a e^{θ} (s i n θ + c o s θ)$ .

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Solution

$x = a e^{θ} (s i n θ - c o s θ)$

$\frac{d x}{d θ} = a e^{θ} (s i n θ - c o s θ) + a e^{θ} (s i n θ + c o s θ)$

$= 2 a e^{θ} s i n θ$
$y = a e^{θ} (s i n θ + c o s θ)$

$\frac{d y}{d θ} = a e^{θ} (s i n θ + c o s θ) + a e^{θ} (c o s θ - s i n θ)$

$= 2 a e^{θ} c o s θ$

$\frac{d y}{d x} = \frac{\frac{d y}{d θ}}{\frac{d x}{d θ}} = \frac{2 a e^{θ} c o s θ}{2 a e^{θ} s i n θ} = c o t θ$

At $x = \frac{π}{4}, \frac{d y}{d x} = c o t \frac{π}{4} = 1$ .

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Similar questions

Q. Find

\frac{d y}{d x}

, when

x = a e^{θ} (\sin θ - \cos θ), y = a e^{θ} (\sin θ + \cos θ)

Q. If

x = e^{θ} (sin θ - cos θ), y e^{θ} (sin θ + cos θ)

, then

\frac{d y}{d x}

θ = \frac{π}{4}

Q. If

x = (cos θ + log tan θ)

and

y = sin θ

,Find the value of

\frac{d y}{d x}

θ = \frac{π}{4}

Q. If

x = a (1 - c o s θ), y = a (θ + s i n θ)

then

\frac{d y}{d x}

θ = \frac{π}{2}

x = sin θ \sqrt{cos 2 θ}, y = cos θ \sqrt{sin 2 θ},

find

\frac{d y}{d x}

θ = \frac{π}{4} .

Insert 0 if the value is not defined and 1 otherwise.

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Find the value of dydx at θ=π4,if x=aeθ(sin θ−cos θ) andy=aeθ(sin θ+cos θ).

x=aeθ(sin θ−cos θ) dxdθ=aeθ(sin θ−cos θ)+aeθ(sin θ+cos θ) =2aeθsinθ y=aeθ(sin θ+cos θ) dydθ=aeθ(sin θ+cos θ)+aeθ(cos θ−sin θ) =2aeθcosθ dydx=dydθdxdθ=2aeθcosθ2aeθsinθ=cot θ At x=π4,dydx=cotπ4=1.

Find the value of $\frac{d y}{d x}$ at $θ = \frac{π}{4}$ ,if $x = a e^{θ} (s i n θ - c o s θ)$ and $y = a e^{θ} (s i n θ + c o s θ)$ .