1.7 Differentiation

• Differentiation is the process of finding the rate of change
• The power rule states that if \( y = ax^n \), then \( \frac{dy}{dx} = nax^{n-1
• Steps to apply the power rule to differentiate a polynomial function

  1️⃣ Identify the coefficient and exponent of each term

  2️⃣ Multiply the coefficient by the exponent

  3️⃣ Reduce the exponent by 1
• What is the coefficient of the term \( x^5 \) in the function \( y = x^5 - 2x^3 + 4 \)?

  1
• For each term in a polynomial function, you must multiply the coefficient by the exponent and then reduce the exponent by 1 when applying the power rule.


  True
• The derivative of $- 5x$ is -5
• What is the derivative of $\sin(x)$?


  $\cos(x)$
• Differentiation allows us to find the gradient of a function at a specific point.


  True
• What is the first step to apply the power rule to a polynomial function?

  Identify coefficient and exponent
• The derivative of -2x^3</latex> is $- 6x^{2}$
• What is the derivative of $3x^{4}$?


  $12x^{3}$
• What is the Chain Rule formula for $y =$$f(g(x))$?


  $\frac{dy}{dx} =$$f'(g(x)) \cdot g'(x)$
• The Product Rule is used to differentiate the product of two functions.


  True
• The Product Rule is applied by adding the product of the first function and the second function's derivative to the product of the first function's derivative and the second function.


  True
• The Product Rule involves adding the derivatives of two functions.

  False
• In the example $y =$$x^{2} *$$sin(x)$, the derivative of $f(x) =$$x^{2}$ is 2x
• Steps to apply the Product Rule for $y =$$x^{2} *$$sin(x)$


  1️⃣ Identify $f(x) =$$x^{2}$ and $g(x) =$$sin(x)$

  2️⃣ Find $f'(x) =$$2x$ and $g'(x) =$$cos(x)$

  3️⃣ Apply the Product Rule formula

  4️⃣ Combine the terms to get $\frac{dy}{dx} =$$(2x * sin(x)) +$$(x^{2} *$$cos(x))$
• The derivative of a function represents its gradient at a specific point.


  True
• What is the derivative of $5x^{3}$ using the Power Rule?


  $15x^{2}$
• The derivative of a constant term is always zero.

  True
• The derivative of $y =$$3x^{4} +$$2x^{2} - 5x +$$7$ is \frac{dy}{dx} = 12x^{3} + 4x - 5</latex>.


  True
• The derivative of $\cos(x)$ is - \sin(x).
• The first step in applying the chain rule is to identify the outer and inner functions.
• What is the derivative of $\cos(3x^{2} +$$1)$?


  $- 6x \sin(3x^{2} +$$1)$
• The Chain Rule formula for differentiating composite functions is \frac{dy}{dx} = f'(g(x)) \cdot g'(x)
• What is the derivative of `y = \cos(3x^2 + 1)` using the Chain Rule?

  `- 6x \sin(3x^2 + 1)`
• What is the formula for the Product Rule?

  $\frac{dy}{dx} =$$f'(x)g(x) +$$f(x)g'(x)$
• The derivative of `y = x^2 * sin(x)` using the Product Rule is $(2x * sin(x)) +$$(x^{2} *$$cos(x))$


  True
• The Quotient Rule is used to differentiate the product of two functions.

  False
• When applying the Quotient Rule, the numerator is $f'(x)g(x) - f(x)g'(x)$


  True
• The chain rule is used to differentiate composite functions.


  True
• What is one key benefit of differentiation?

  Finding tangent slopes
• What is the derivative of \( y = 5x^3 \) using the power rule?

  15x^2
• What is the derivative of \( y = 3x^4 + 2x^2 - 5x + 7 \)?

  12x^3 + 4x - 5
• What should you do with the exponent when applying the power rule in differentiation?

  Reduce it by 1
• What is the derivative of $3x^{4}$ using the power rule?


  $12x^{3}$
• What is the derivative of the constant term 4?

  $0$
• The derivative of $\cos(x)$ is -\sin(x)</latex>
• What is one application of differentiation in physics?

  Calculating velocity
• The derivative of $3x^{4} +$$2x^{2} - 5x +$$7$ is $12x^{3} +$$4x - 5$.


  True