Differentiation — Quiz 1

Questions in this quiz (10)

1. If $$y=3x^3-4x+5$$, find $$y'(1)$$ and $$y''(1)$$.

   • $$y'(1)=5$$, $$y''(1)=18$$
   • $$y'(1)=9,\ y''(1)=18$$
   • $$y'(1)=5,\ y''(1)=12$$
   • $$y'(1)=1,\ y''(1)=18$$

2. If $$f(x)=2x^3+5x-1$$, find $$f'(2)$$.

   • $$31$$
   • $$27$$
   • $$23$$
   • $$29$$

3. If $$y=\sqrt{4-x^2}$$, find $$\frac{dy}{dx}$$.

   • $$-\frac{2x}{\sqrt{4-x^2}}$$
   • $$-\frac{x}{\sqrt{4-x^2}}$$
   • $$-\frac{1}{\sqrt{4-x^2}}$$
   • $$\frac{x}{\sqrt{4-x^2}}$$

4. If $$y=\sqrt{1-4x^2}$$, find $$\frac{dy}{dx}$$.

   • $$-\frac{1}{\sqrt{1-4x^2}}$$
   • $$-\frac{4x}{\sqrt{1-4x^2}}$$
   • $$-\frac{2x}{\sqrt{1-4x^2}}$$
   • $$\frac{4x}{\sqrt{1-4x^2}}$$

5. If $$h(t)=(t^3+2t)(t-3)$$, find $$h'(t)$$.

   • $$4t^3-6t^2+4t-6$$
   • $$3t^3-9t^2+4t-6$$
   • $$4t^3-9t^2+2t-6$$
   • $$4t^3-9t^2+4t-6$$

6. If $$f(t)=e^{2t}\sin(2t)$$, find $$f'(t)$$.

   • $$2e^{t}(\sin(2t)+\cos(2t))$$
   • $$e^{2t}(2\sin(2t)+\cos(2t))$$
   • $$2e^{2t}(\sin(2t)-\cos(2t))$$
   • $$2e^{2t}(\sin(2t)+\cos(2t))$$

7. If $$g(x)=\frac{x^3}{x+1}$$, find $$g'(x)$$.

   • $$\frac{x^2(2x-3)}{(x+1)^2}$$
   • $$\frac{2x^3+3x}{(x+1)^2}$$
   • $$\frac{x^2(3x+2)}{(x+1)^2}$$
   • $$\frac{x^2(2x+3)}{(x+1)^2}$$

8. If $$h(x)=\frac{\cos x}{x}$$, find $$h'(x)$$.

   • $$\frac{-\sin x-\cos x}{x}$$
   • $$\frac{-x\cos x-\sin x}{x^2}$$
   • $$\frac{x\sin x-\cos x}{x^2}$$
   • $$\frac{-x\sin x-\cos x}{x^2}$$

9. If $$y=\tan(3x^2+2)$$, find $$\frac{dy}{dx}$$.

   • $$3x\sec^2(3x^2+2)$$
   • $$6x\sec(3x^2+2)$$
   • $$6x\sec^2(3x^2+2)$$
   • $$6\sec^2(3x^2+2)$$

10. If $$x=\cos^3(t-2)$$, find $$\frac{dx}{dt}$$.

    • $$-3\cos^2(t-2)\sin(t-2)$$
    • $$-\cos^3(t-2)$$
    • $$-3\cos(t-2)\sin^2(t-2)$$
    • $$3\cos^2(t-2)\sin(t-2)$$