Find d y d x if : x= ( t), y= ( t) - Vidyadip

Question

Find $\frac{d y}{d x}$ if : $x=\cos (\log t), y=\log (\cos t)$

✓

Answer

Given, $y=\log (\cos t)$
Differentiate $w . r . t . t$
$
\frac{d y}{d t}=\frac{d}{d t}[\log (\cos t)]=\frac{1}{\cot t} \cdot \frac{d}{d t}(\cos t)=\frac{1}{\cot t}(-\sin t) \quad \therefore \frac{d y}{d t}=-\tan t
$
And, $x=\cos (\log t)$
Differentiate w.r.t.t
$
\frac{d x}{d t}=\frac{d}{d t}[\cos (\log t)]=-\sin (\log t) \cdot \frac{d}{d t}(\log t)=-\frac{\sin (\log t)}{t} \quad \therefore \frac{d x}{d t}=-\frac{\sin (\log t)}{t} .
$
Now, $\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{-\tan t}{-\frac{\sin (\log t)}{t}} \ldots[$ From (I) and (II) ]
$
\therefore \quad \frac{d y}{d x}=\frac{t \cdot \tan t^t}{\sin (\log t)}
$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(2 Marks)" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

A bag contains 7 red, 5 white and 8 black balls. If four balls are drawn one by one with replacement, what is the probability that
none is white?

Find the cofactors of the elements for the matrix $\left[\begin{array}{ll}-1 & 2 \\ -3 & 4\end{array}\right]$

If $\vec{\text{a}},\vec{\text{b}}$ are two vectors, then write the truth value of the following statement:$\big|\vec{\text{a}}\big|=\big|\vec{\text{b}}\big|\Rightarrow\vec{\text{a}}=\vec{\text{b}}$

Write a value of $\int\sqrt{\text{x}^2-9}\text{ dx}$

Find the values of x for which the following functions are strictly decreasiong:

$f(x)=x^3-9 x^2+24 x+12$

Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=2\text{x}^2-5\text{x}+3\text{ on }[1,3]$

Show that each of the following equation represents a pair of lines : $x^2-2 \sqrt{3} x y-y^2=0$

Evaluate the following:
$\sin^{-1}\Big(\sin\frac{17\pi}{8}\Big)$

If statements p, q are true and r, s are false, determine the truth values of the following.

(p ∧ ~r) ∧ (~q ∨ s)

If $\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\\\text{z}\end{bmatrix}=\begin{bmatrix}1\\-1\\0\end{bmatrix}$, find x, y and z.