Differentiation...

$$\displaystyle \frac{d}{dx}\left ( \tan ^{-1}\left ( \frac{\sqrt{x}-x}{1+x^{3/2}} \right ) \right )$$ equals $$\displaystyle ($$for $$x\geq 0)$$

The area of the triangle whose vertices are $$A(1,1), B(7, 3)$$ and $$C(12, 2)$$ is

$$\displaystyle \frac{d}{dx}\left ( \tan ^{-1}\left ( \frac{a-x}{1+ax} \right ) \right )$$ equals if ax > -1

If $$f(x) = \displaystyle \left | \cos x-\sin x \right |$$ then $$\displaystyle f'\left ( \dfrac{\pi}4 \right )$$ is equal to-

If $$\displaystyle y=\frac{1}{1+x^{\beta -\alpha}+x^{\gamma -\alpha}}+\frac{1}{1+x^{\alpha-\beta}+x^{\gamma -\beta }}+\frac{1}{1+x^{\alpha -\gamma }+x^{\beta-\gamma }}$$
then $$\displaystyle \frac{dy}{dx}$$ is equal to-

If $$\displaystyle y=\left | \cos x \right |+\left | \sin x \right |$$ then $$\displaystyle \frac{dy}{dx}$$ at $$x=\dfrac{2\pi }{3}$$ is:

$$\displaystyle \frac{d}{d\theta }\left ( \tan ^{-1}\left ( \frac{1-\cos \theta }{\sin \theta } \right ) \right )$$ equals if $$\displaystyle-\pi <\theta <\pi $$

Find the area of the right-angled triangle whose vertices are $$(2, -2)$$ , $$(-2, 1)$$ and $$(5, 2).$$

Find the area of the triangle whose vertices are $$(a, b + c), (a, b - c)$$ and $$(-a, c)$$.

Let $$\displaystyle y=(1+x^{2})\tan^{-1}(x-x)$$ and $$\displaystyle f(x)=\frac1{2x}\frac {dy}{dx},$$ then $$f(x)+\cot^{-1}x$$ is equal to