Question

Find $\frac{dy}{dx}$, if x = sec² θ, y = tan³ θ, at θ = π/3

Answer 1

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Answer 2

Answer:

2.598.

Step-by-step explanation:

Given:

• x = sec² θ.

• y = tan³ θ.

We know,  

$\rm \dfrac {dy}{dx} = \dfrac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}.$

Now,

$\rm \dfrac{dy}{d\theta} = \dfrac{d(\tan^3\theta)}{d\theta} \\=3\tan^2\theta\dfrac{d(\tan\theta)}{d\theta} \\=3\tan^2\theta\sec^2\theta.\\\\\\\dfrac{dx}{d\theta} = \dfrac{d(\sec^2\theta)}{d\theta} \\=2\sec\theta\dfrac{d(\sec\theta)}{d\theta} \\=2\sec\theta(\sec\theta\tan\theta).\\=2\sec^2\theta\tan\theta.$

Therefore,

$\rm \dfrac{dy}{dx}=\dfrac{\dfrac{dy}{d\theta}}{\dfrac{dx}{d\theta}}=\dfrac{3\tan^2\theta\sec^2\theta}{2\sec^2\theta\tan\theta}=\dfrac 32 \tan\theta.$

At $\theta = \dfrac \pi 3$,

$\rm \dfrac{dy}{dx}=\dfrac 32 \tan\left ( \dfrac \pi 3 \right )=2.598.$