Differentiate the following functions with respect to x: ( )^ - Vidyadip

Question

Differentiate the following functions with respect to x:
$(\log\text{x})^{\cos\text{x}}$

✓

Answer

Let $\text{y}=(\log\text{x})^{\cos\text{x}}\ .....(\text{i})$
Taking log on both the sides,
$\log\text{y}=(\log\text{x})^{\cos\text{x}}$
$\Rightarrow\log\text{y}=\cos\text{x}\log(\log\text{x})$
Differentiating with respect to x,
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\cos\text{x}\frac{\text{d}}{\text{dx}}\log(\log\text{x})+\log(\log\text{x})\frac{\text{d}}{\text{dx}}(\cos\text{x})$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{\cos\text{x}}{\log\text{x}}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log(\log\text{x})\times(-\sin\text{x})$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{\cos\text{x}}{\log\text{x}}\times\big(\frac{1}{\text{x}}\big)-\sin\text{x}\log(\log\text{x})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}\Big[\frac{\cos\text{x}}{\text{x}\log\text{x}}-\sin\text{x}\log(\log\text{x})\Big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=(\log\text{x}^{\cos\text{x}})\Big[\frac{\cos\text{x}}{\text{x}\log\text{x}}-\sin\text{x}\log(\log\text{x})\Big]$
[Using equation (i)]

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 "4 Marks" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove that:
$\begin{vmatrix}\text{a}^2+1&\text{ab}&\text{ac}\\\text{ab}&\text{b}^2+1&\text{bc}\\\text{ca}&\text{cb}&\text{c}^2+1 \end{vmatrix}=1+\text{a}^2+\text{b}^2+\text{c}^2$

How many of circuits of Type A and of Type B, should be produced by the manufacturer so as to maximise his profit? Determine the maximum profit.

Evaluate $\int \frac{x^2 d x}{(x \sin x+\cos x)^2}$.

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
f(x) = x(x - 1) on [1, 2]

Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b - ab, for all a, b ∈ S.
Prove that:

* is a binary operation on S.
* is commutative as well as associative.

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{x}}{1+6\text{x}^3}\Big)$

Compute the adjoint of the following matrices:

$\begin{bmatrix}2 & 0 & -1 \\5 & 1 & 0 \\ 1 & 1 & 3 \end{bmatrix}$

Verify that (adj A)A = |A| I = A (adj A) for the above matrices.

An insurance company insured 2000 scooters and 3000 motorcycles. The probability of an accident involving a scooter is 0.01 and that of a motorcy is 0.02. An insured vehicle met with an accident. Find the probability that the accidented vehicle was a motorcycle.

Find the equation of the plane which bisects the line segment joining the points (-1, 2, 3) and (3, -5, 6) at right angles.

Solve the following system of equations by matrix method:
6x - 12y + 25z = 4
4x + 15y - 20z = 3
2x + 18y + 15z = 10