Exercise Set 10

1. Find \(\dfrac{dy}{dx}\) for the following by implicit differentiation.

   1. \(x^2y+3xy^3-x=3\)
   2. \(\sin(x^2y^2)=x\)
   3. \(xy^2=y+e^{xy}\)

2. Consider the curve \(x^2+3xy+y^2+2x-7=0\). Use implicit differentiation to find \(\dfrac{dy}{dx}\) and hence compute the equation of the tangent at the point \((1,1)\).

3. Find the slope of the circle \(x^2+y^2=25\) at the point \((-3,-4)\). Hence find the equations of the tangent and normal at that point.

4. Find \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\) for the following by implicit differentiation.

   1. \(y+\sin(y)=x\)
   2. \(x^2-xy+y^2=3\)

5. Find \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\) for the following parametric equations.

   1. \(y=\cos(2t)\) and \(x=\sin(t)\)
   2. \(y=\dfrac{3+2t}{1+t}\) and \(x=\dfrac{2-3t}{1+t}\)
   3. \(y=3\sin(\theta)-\sin^3(\theta)\) and \(x=\cos^3(\theta)\)

6. Find the derivative of the following inverse functions by implicit differentiation.

   1. \(y=\sin^{-1}(x)\)
   2. \(y=\cosh^{-1}(3x)\)
   3. \(y=\tan^{-1}(4x^2)\)

7. Find \(\dfrac{dy}{dx}\) for the following using logarithmic differentiation.

   1. \(y=a^x\)
   2. \(y=\dfrac{(x-4)^7(2x+3)^2}{(4x+7)^3}\)

8. Find \(\dfrac{dy}{dx}\) for the following.

   1. \(y=x^x\)
   2. \(y=(\tanh(x))^x\)
   3. \(x^3+\sin(xy)=xy^2\)