Chain Rule

Chain Rule on Differentiation

Multiple Choice

Which one is the same as

 $\left(f\ \circ\ g\right)\left(x\right)$  

•  $g\left(f\left(x\right)\right)$  

•  $f\left(g\left(x\right)\right)$  
  

Multiple Choice

Given  $f\left(x\right)=2x$  and
 $g(x)=x^2+3$   , find  $\left(f\ \circ\ g\right)\left(x\right)$ .

•  $x^2+2x+3$  

•  $4x^2+3$  

•  $2x^2+3$  

•  $2x^2+6$  

Multiple Choice

Given  $f\left(x\right)=3x+10$   
and  $g\left(x\right)=x-2\$ .
Find  $\left(f\ \circ\ g\right)\left(5\right)$  

• 19

• 23

• -10

• None of these.

Multiple Choice

find  $y'$ for   $y=\left(2x+1\right)^{10}$  

• 10(2x+1)⁹

• 20(2x-1)⁹

• 20(2x+1)¹⁰

• 20(2x+1)⁹

Multiple Choice

Find the derivative of f(x) = (x⁶ + 4)⁵

• f '(x) = 5x⁵(x⁴ + 4)⁴

• f '(x) = 6x⁵(x⁶ + 4)⁴

• f '(x) = 30x⁵(x⁶ + 4)⁴

• f '(x) = 30x⁶(x⁶ + 4)⁴

Multiple Choice

Find the derivative
f(t) = (t² + 2t)⁵

• f'(t) = 5(2t+2)⁴
• f'(t) = 5(t² + 2t)⁴
•  f'(t) = 5(t² + 2t)⁴(2t)
• f'(t) = 5(t² + 2t)⁴(2t + 2)

Multiple Choice

Given  $F\left(x\right)=\left(\frac{x}{2x+1}\right)^3$ ,    find  $F'\left(x\right)$ . 

•  $3\left(\frac{x}{2x+1}\right)^2$  

•  $3\left[\frac{1}{\left(2x+1\right)^2}\right]^2$  

•  $\frac{1}{2}$  

•  $\frac{3x^2}{\left(2x+1\right)^4}$  

Multiple Choice

Differentiate $f\left(x\right)=\sqrt{x^2-1}$ . 

•  $f'\left(x\right)=\sqrt{x^2-1}$  
  

•  $f'\left(x\right)=\frac{1}{2x\sqrt{x^2-1}}$  
  

•  $f'\left(x\right)=\frac{x}{\sqrt{x^2-1}}$  

•  $f'\left(x\right)=x\sqrt{x^2-1}$