The Chain Rule

Chain Rule on Differentiation

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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

$$y=\left(2x^2+3x+4\right)^{\frac{1}{2}}$$  Differentiate using the chain rule

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

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

Multiple Choice

$$y\ =\sin\left(3x^2+4\right)$$  Differentiate

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

• $$\ 6x\cos\left(3x^2\ +4\right)$$  
  

Multiple Choice

find y' for y= sin(2x²)

• 4sin(2x²)

• 4xcos(2x²)

• 2xcos(2x²)

• xcos(4x²)

Multiple Choice

Multiple Choice

Differentiate $$\left(x+3\right)^5$$  

• $$5\left(x+3\right)^4$$  

• $$15\left(x+3\right)^4$$  

• $$5x\left(x+3\right)^4$$  

• $$15x\left(x+3\right)^4$$  

Multiple Choice

Differentiate $$\left(2x-1\right)^3$$  

• $$3\left(2x-1\right)^2$$  

• $$6\left(2x-1\right)^2$$  

• $$6x\left(2x-1\right)^2$$  

• $$2\left(2x-1\right)^3$$  

Multiple Choice

Differentiate $$\left(8-x\right)^7$$  

• $$7\left(8-x\right)^6$$  

• $$-7\left(8-x\right)^6$$  

• $$-7x\left(8-x\right)^6$$  

• $$7\left(8-x\right)^7$$  

Multiple Choice

Differentiate $$2\left(3x+4\right)^6$$  

• $$36\left(3x+4\right)^5$$  

• $$6\left(3x+4\right)^5$$  

• $$12\left(3x+4\right)^5$$  

• $$12x\left(3x+4\right)^5$$  

Multiple Choice

Differentiate $$\left(6-5x\right)^4$$  

• $$20\left(6-5x\right)^3$$  

• $$4\left(6-5x\right)^3$$  

• $$-20\left(6-5x\right)^3$$  

• $$-5\left(6-5x\right)^3$$  

Multiple Choice

Differentiate $$\frac{1}{x-2}$$  

• $$-\left(x-2\right)^{-2}$$  

• $$\left(x-2\right)^{-2}$$  

• $$-2\left(x-2\right)^{-2}$$  

• $$2\left(x-2\right)^{-2}$$  

Multiple Choice

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

• $$-24\left(2x+3\right)^{-4}$$

• $$-12\left(2x+3\right)^{-4}$$

• $$24\left(2x+3\right)^{-4}$$  

• $$-12\left(2x+3\right)^{-4}$$

Multiple Choice

Differentiate $$\sqrt[]{4x-1}$$  

• $$2(4x−1)^{-\frac{1}{2}}$$  

• $$2(4x−1)^{\frac{1}{2}}$$  

• $$4(4x−1)^{-\frac{1}{2}}$$  

• $$4(4x−1)^{\frac{1}{2}}$$  

Multiple Choice

Find the derivative f(x) = (x² + 2x)⁵

• f'(x) = 5(2x+2)⁴
• f'(x) = 5(x² + 2x)⁴
•  f'(x) = 5(x² + 2x)⁴(2x)
• f'(x) = 5(x² + 2x)⁴(2x + 2)

Multiple Choice

Find $$y'$$ if  $$y=-3\left(2x-5\right)^4$$   

• $$y'=-24\left(2x-5\right)^3$$  

• $$y'=-12\left(2x-5\right)^3$$  

• $$y'=-24\left(2x-5\right)^4$$  

• $$y'=-12\left(2x-5\right)^4$$  

Multiple Choice

Find the derivative of the given equation
f(x) = x³ + x² + 3

• 3x² + 2x
• 3x + 2x 
• 3x + 2x + 3
• x³ + x² 

Multiple Choice

$$f\left(x\right)=\sqrt{x^2+8}$$  
Which statement is true?

• $$\sqrt{x}$$  is the outside function,  $$x^2+8$$  is the inside function. 

• $$x^2+8$$  is the outside function,  $$\sqrt{x}$$  is the inside function. 

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

What is the derivative of $$f(x) = (3x^2 + 4x + 5)^6$$ ?

• $$6(3x^2 + 4x + 5)^5$$

• $$6(3x^2 + 4x + 5)^5 \cdot (6x + 4)$$

• $$(6x + 4)^5$$

• $$18x + 4$$

Multiple Choice

What is the derivative of $$f(x) = (2x^3 - x)^4$$ ?

• $$4(2x^3 - x)^3$$

• $$12x^2(2x^3 - x)^3$$

• $$4(2x^3 - x)^3 \cdot (6x^2 - 1)$$

• $$(6x^2 - 1)^4$$