Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Differentiate w.r.t. x the function in Exercise:
$\text{x}^\text{x}+\text{x}^\text{a}+\text{a}^\text{x}+\text{a}^\text{a},$ for some fixed $a > 0$ and $x > 0$
✓

Answer

Let $\text{y}=\text{x}^\text{x}+\text{x}^\text{a}+\text{a}^\text{x}+\text{a}^\text{a}$Also, let $x^x = u, x^a = v, a^x = w$, and $a^a = s$
$\therefore\ $ y = u + v + w + s
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{du}}{\text{dx}}+\frac{\text{dv}}{\text{dx}}+\frac{\text{dw}}{\text{dx}}+\frac{\text{ds}}{\text{dx}}$
$U = x^x$
$\Rightarrow\ \log\text{u}=\log\text{x}^\text{x}$
$\Rightarrow\ \log\text{u}=\text{x}\log\text{x}$
Differentiating both sides with respect to x, we obtain
$\frac{1}{\text{u}}\frac{\text{du}}{\text{dx}}=\log\text{x}.\frac{\text{d}}{\text{dx}}(\text{x)}+\text{x}.\frac{\text{d}}{\text{dx}}(\log\text{x})$
$\Rightarrow\ \frac{\text{du}}{\text{dx}}=\text{u}\Big[\log\text{x}.1+\text{x}.\frac{1}{\text{x}}\Big]$
$V = x^a$
$\therefore\ \frac{\text{dv}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(\text{x}^\text{a})$
$\Rightarrow\ \frac{\text{dv}}{\text{dx}}=\text{ax}^{\text{a}-1}\ \dots(3)$
$W = a^x$
$\Rightarrow\ \log\text{w}=\log\text{a}^\text{x}$
$\Rightarrow\ \log\text{w}=\text{x}\log\text{a}$
Differentiating both sides with respect to x, we obtain
$\Rightarrow\ \frac{\text{dw}}{\text{dx}}=\text{w}\log\text{a}$
$\Rightarrow\ \frac{\text{dw}}{\text{dx}}=\text{a}^\text{x}\log\text{a}\ \dots(4)$
$S = a^a$
Since a is constant, $a^a$ is also a constant.
$\therefore\ \frac{\text{ds}}{\text{dx}}=0\ \dots(5)$
From (1), (2), (3), (4), and (5), we obtain
$\frac{\text{dy}}{\text{dx}}=\text{x}^\text{x}(1+\log\text{x})+\text{ax}^{\text{a}-1}+\text{a}^\text{x}\log\text{a}+0$

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 "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following system of equations by matrix method:
$x + y + z = 6$
$x + 2z = 7$
$3x + y + z = 12$
→
An article manufactured by a company consists of two parts X and Y. In the process of manufacture of the part X, 9 out of 100 parts may be defective. Similarly, 5 out of 100 are likely to be defective in the manufacture of part Y. Calculate the probability that the assembled product will not be defective.
→
For each of the differential equations given in find a particular solution satisfying the given condition:
$\frac{\text{dy}}{\text{dx}}+2\text{y}\tan\text{x}=\sin\text{x};\text{y}=0\ \text{when x}=\frac{\pi}{3}$
→
For the following differntial equations verify that the accompanying function is a solution:
Differential equation Function
$\text{y}=\Big(\frac{\text{dy}}{\text{dx}}\Big)^2$ $\text{y}=\frac{1}{4}(\text{x}\pm\text{a})^2$
→
Find the equation of the plane which contains planes is the line of intersection of the planes x + 2y + 3z - 4 = 0 and 2x + y - z + 5 = 0 and whose x-intercept is twice its z-intercept.
→
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 - 1)(x - 2)(x - 3) on [0, 4]
→
Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{4}}\sec\text{x}\text{ dx}$
→
Differentiate $\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)$ with respect to $\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big),$ if 0 < x < 1.
→
Form the differential equation of the family of curves represented by the equation (a being the perimeter):$(\text{x}-\text{a})^2+2\text{y}^2=\text{a}^2$
→
Show that the differential equation $2 y e^{\frac{x}{y}} d x+\left(y-2 x e^{\frac{x}{y}}\right) d y=0$ is homogeneous and find its particular solution, given that, x = 0 when y = 1.
→