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

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d bloggers like this: