Chain rule Quiz 1

Question 1

Find $$\dfrac{d}{dx}\left( \sec \left( {{e}^{2x}} \right) \right)$$.

Accepted Answer

$$2{{e}^{2x}}\sec \left( {{e}^{2x}} ight)	an \left( {{e}^{2x}} ight)$$

Answer

$$\sec \left( {{e}^{2x}} ight)	an \left( {{e}^{2x}} ight)$$

Answer

$${{e}^{2x}}\sec \left( {{e}^{2x}} ight)	an \left( {{e}^{2x}} ight)$$

Answer

none of the above

Question 2

Find the sum of $$\dfrac{d}{dx}\left( {{\sin }^{2}}\left( {{e}^{x}} \right) \right)$$ and $$\dfrac{d}{dx}\left( {{e}^{{{\cos }^{2}}x}} \right)$$

Accepted Answer

$${{e}^{x}}\sin \left( 2{{e}^{x}} \right)-{{e}^{{{\cos }^{2}}x}}\sin \left( 2x \right)$$

Answer

$$2\sin \left( {{e}^{x}} \right)\cos \left( {{e}^{x}} \right)-{{e}^{{{\cos }^{2}}x}}\sin \left( x \right)$$

Answer

$${{e}^{x}}\sin \left( {{e}^{2x}} \right)-{{e}^{{{\cos }^{2}}x}}\sin \left( 2x \right)$$

Answer

$$2{{e}^{x}}\sin \left( 2{{e}^{x}} \right)-{{e}^{{{\cos }^{2}}x}}\cos \left( x \right)$$

Question 3

Find the difference of $$\dfrac{d}{dx}\left( {{e}^{{{	an }^{2}}x}} ight)$$ and $$\dfrac{d}{dx}\left( {{	an }^{-1}}\left( \sec x ight) ight)$$.

Accepted Answer

$$2	an x{{\sec }^{2}}x.{{e}^{{{	an }^{2}}x}}-\dfrac{\sec x	an x}{1+{{\sec }^{2}}x}$$

Answer

$$2	an x{{\sec }^{2}}x.{{e}^{{{	an }^{2}}x}}-\dfrac{\sec x	an x}{2+{{\sec }^{2}}x}$$

Answer

$$2{{\sec }^{2}}x.{{e}^{{{	an }^{2}}x}}-\dfrac{\sec x}{2+{{\sec }^{2}}x}$$

Answer

$$2{{\sec }^{2}}x.{{e}^{{{	an }^{2}}x}}-\dfrac{	an x}{2+{{\sec }^{2}}x}$$

Question 4

Find the ratio of $$\dfrac{d}{dx}\left( {{	an }^{-1}}\left( x ight) ight)$$ and $$\dfrac{d}{dx}\left( {{	an }^{-1}}\left( 2x ight) ight)$$

Accepted Answer

$$\dfrac{1+4{{x}^{2}}}{2\left( 1+{{x}^{2}} \right)}$$

Answer

$$\dfrac{1+4{{x}^{2}}}{1+{{x}^{2}}}$$

Answer

$$\dfrac{1+{{x}^{2}}}{1+4{{x}^{2}}}$$

Answer

$$\dfrac{2\left( 1+{{x}^{2}} \right)}{1+4{{x}^{2}}}$$

Question 5

Evaluate $$\dfrac{\dfrac{d}{dx}\left( \ln \left( x-1 \right) \right)-\dfrac{d}{dx}\left( \ln \left( x \right) \right)}{\dfrac{d}{dx}\left( \ln \left( x-1 \right) \right).\dfrac{d}{dx}\left( \ln \left( x \right) \right)}$$.

Accepted Answer

$$1$$

Answer

$$2$$

Answer

$$3$$

Answer

$$4$$