## Derivatives: part 1: IITJEE Maths Tutorial Problems Practice

Problem 1:

If $y=x^{x}$, $x>0$, then find $\frac{dy}{dx}$.

Problem 2:

If $y= x^{x^{x^{\ldots}}}$, then find the value of $x\frac{dy}{dx}$.

Problem 3:

Find the derivative of $e^{\ln{x}}$ w.r.t. x.

Problem 4:

Let $f(x) = \log{(x+\sqrt{x^{2}+1})}$, then find the value of $f^{'}(x)$.

Problem 5:

If $y= \arctan{\frac{\sqrt{1+x^{2}}-1}{x}}$, then find the value of $y^{'}(0)$.

Problem 6:

If $y=t^{2}+t-1$, then find the value of $\frac{dy}{dx}$.

Problem 7:

If $x=a(t-\sin{t})$, $y=a(1+\cos{t})$, then evaluate $\frac{dy}{dx}$.

Problem 8:

If $x^{y}=e^{x-y}$, then evaluate $\frac{dy}{dx}$.

Problem 9:

If $y= \sec^{-1}{(\frac{x+1}{x-1})} + \arcsin{(\frac{x-1}{x+1})}$, then evaluate $\frac{dy}{dx}$.

Problem 10:

If $y = \arctan{(\frac{\sin{x}+\cos{x}}{\cos{x}-\sin{x}})}$, then find $\frac{dy}{dx}$

Problem 11:

If $\sqrt{x}+\sqrt{y}=4$, then evaluate $\frac{dy}{dx}$ at $y=1$.

Problem 12:

If $f(x) = \frac{x-4}{2\sqrt{x}}$, then evaluate $f^{'}(0)$.

Problem 13:

If $f^{'}(x) = \sin{\log{x}}$ and $y=f(\frac{2x+3}{3-2x})$, find $\frac{dy}{dx}$. One of the given choices is correct:

(a) $\frac{12\cos{(\log{x})}}{x(3-2x)^{2}}$

(b) $\frac{12\sin{\log{(\frac{2x+3}{3-2x})}}}{(3-2x)^{2}}$

(c) $\frac{12\cos{\log{(\frac{2x+3}{3-2x})}}}{x(3-2x)^{2}}$

(d) none of these

Problem 14:

If $f(0)=0=g(0)$ and $f^{'}(0)=6=g^{'}(0)$, then $\lim_{x \rightarrow 0} \frac{f(x)}{g(x)}$ is given by:

(a) 1 (b) 0 (c) 12 (d) -1

Regards,

Nalin Pithwa

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