Derivatives. Chain rule

Mastering the Chain Rule

Mastering the Chain Rule

The chain rule is a powerful tool in calculus that allows us to find the derivative of composite functions.

• Helps find derivative of outer and inner functions

• Product of outer function's derivative and inner function's derivative

• Solves complex differentiation problems

• Understanding and applying chain rule essential

• Unleashes power of derivatives

Multiple Choice

What is the purpose of the chain rule in calculus?

• To find the derivative of composite functions

• To solve complex integration problems

• To simplify algebraic expressions

• To calculate the limit of a function

Chain Rule: Derivative of Composite Functions

The chain rule is a fundamental concept in calculus that allows us to find the derivative of composite functions. It is used to differentiate functions that are composed of other functions. By applying the chain rule, we can break down complex functions into simpler parts and find their derivatives. This rule is essential in various fields of mathematics and science, including physics and engineering.
Understanding composite functions and how to unleash the power of derivatives. Learn how to differentiate complex functions by breaking them down into simpler components. Apply the chain rule to find the derivative of composite functions.

Multiple Choice

What is the purpose of applying the chain rule to find the derivative of composite functions?

• To simplify complex functions into simpler components

• To understand the power of derivatives

• To differentiate composite functions

• To master the chain rule

• STEP 1) INDENTIFY A FUNCTION WITHIN A FUNCTION: Identify which is f(x) and which is g(x) and how they are composed f(g(x)).

• STEP 2) BREAK IT DOWN: Instead of tackling it all at once, break it into parts.

  • First, find the derivative of the outer function f', treating g(x) as just a variable.

  • Then, find the derivative of the inner function g', which is the derivative of g(x).

• STEP 3) MULTIPLY THEM: Now that you have the derivatives of both outer and inner functions, multiply them together, which will give the derivative of the whole composite function, f(g(x)).

Carefully watch the video and try to understand the chain rule.
Follow the steps:

Multiple Choice

What is the purpose of the chain rule in calculus?

• To find the derivative of composite functions

• To simplify complex functions

• To solve equations involving derivatives

• To evaluate limits of functions

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

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

Differentiation is also known as

• Comparison

• Integration

• Derivatives

• Multiplication

Multiple Choice

Derivative means the same thing as

• slope of the tangent line
• slope of the normal line
• exponent
• warning

Multiple Choice

What rule should be used in deriving $$h\left(x\right)=2\left(x-4\right)^3$$

• Chain rule

• Product rule

• Quotient rule

• Sum rule

Multiple Choice

When do you use the chain rule?

• anytime you want
• when there is a function in a function (composition of functions)

• when there is division of two functions

• when there is product of two functions

Multiple Choice

What rule will apply to find the derivative of $$y=x\ln x$$

• Product rule

• Quotient rule

• Power rule

• Sum rule

Multiple Choice

What rule should be used in deriving $$f\left(x\right)=x^5$$

• Chain rule

• Quotient rule

• Power rule

• Product rule

Multiple Choice

When do you use the quotient rule?

• anytime you want

• when there is a function in a function (composition of functions)

• when there is division of two functions

• when there is product of two functions

Multiple Choice

Find the derivative

$$f\left(x\right)=3x^3-9x^2+11x-2$$  

• $$f'\left(x\right)=9x^2-18x+11$$  

• $$f'\left(x\right)=9x^3-18x^2+11$$  

• $$f'\left(x\right)=3x^2-9x+11$$  

• $$f'\left(x\right)=9x-18$$  

Multiple Choice

Use the product rule to find the derivative of
$$f\left(x\right)=x^2\sin x$$  

• $$f'\left(x\right)=2x\cos x$$  

• $$f'\left(x\right)=2x\sin x+x^2\cos x$$  

• $$f'\left(x\right)=2x\sin x-x^2\cos x$$  

• $$f'\left(x\right)=x\sin x+x^2\cos x$$  

Multiple Choice

Use any method you want to find the derivative of
$$g\left(x\right)=\left(2x-3\right)\left(x+9\right)$$  

• $$g'\left(x\right)=2x^2+15x-27$$  

• $$g'\left(x\right)=4x+15$$  

• $$g'\left(x\right)=2$$  

• $$g'\left(x\right)=2x$$  

Multiple Choice

Use division rule to find the derivative of
$$h\left(x\right)=\frac{x^3-9x+4}{x}$$  

• $$h'\left(x\right)=x^2-9+4x^{-1}$$  

• $$h'\left(x\right)=2x-\frac{4}{x^2}$$  

• $$h'\left(x\right)=2x+\frac{4}{x^2}$$  

• $$h'\left(x\right)=2x-\frac{4}{x}$$  

Multiple Choice

Differentiate:
$$y=x^{\frac{2}{3}}+4x^{\frac{1}{4}}$$  

• $$\frac{dy}{dx}=\frac{2}{3x^{\frac{1}{3}}}+\frac{1}{x^{\frac{3}{4}}}$$  

• $$\frac{dy}{dx}=-\frac{2}{3x^{\frac{1}{3}}}+\frac{1}{x^{\frac{3}{4}}}$$  

• $$\frac{dy}{dx}=\frac{2}{3x^{\frac{1}{3}}}-\frac{1}{x^{\frac{3}{4}}}$$  

• $$\frac{dy}{dx}=-\frac{2}{3x^{\frac{1}{3}}}-\frac{1}{x^{\frac{3}{4}}}$$  

Multiple Choice

Find the derivative:
$$f\left(x\right)=\frac{1}{x}-\frac{1}{x^2}+\frac{1}{x^3}$$  

• $$f'\left(x\right)=x^{-1}-x^{-2}+x^{-3}$$  

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

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

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

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 of

$$f\left(t\right)=\left(t^2+2t\right)^5$$

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

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

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

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

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

Multiple Choice

Differentiate $$f\left(x\right)=\frac{1}{x^2-3}$$ . 

• $$f'\left(x\right)=2x-3$$  
  

• $$f'\left(x\right)=\frac{1}{2x-3}$$  
  

• $$f'\left(x\right)=\frac{-2x}{\left(x^2-3\right)^2}$$  

• $$f'\left(x\right)=\frac{2x}{\left(x^2-3\right)^2}$$  

Multiple Choice

Differentiate $$f\left(x\right)=\ln\left(x^2-3\right)$$ . 

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

• $$f'\left(x\right)=\frac{2x}{2x-3}$$  
  

• $$f'\left(x\right)=\frac{2x}{x^2-3}$$  

• $$f'\left(x\right)=\frac{2x}{\left(x^2-3\right)^2}$$  

Multiple Choice

Differentiate $$f\left(x\right)=2^{x^2-3}$$ . 

• $$f'\left(x\right)=2^{x^2-3}\ ·\ 2x$$  
  

• $$f'\left(x\right)=2^{x^2-3}\ ·\ \ln2\ ·\ 2x$$  
  

• $$f'\left(x\right)=2^{x^2-3}$$  

• $$f'\left(x\right)=2^{x^2-3}\ ·\ \ln2$$