3.8 G: Differentiation

• The derivative represents the slope of the tangent line to the function's graph at a particular point.


  True
• What does a positive derivative indicate about the function's behavior?

  Function is increasing
• The derivative is the instantaneous rate of change of a function.


  True
• The power rule states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).


  True
• What is a polynomial function defined as?

  Sum of terms with coefficients
• The derivative of any constant function is always zero.

  True
• What is the derivative of \( f(x) = x^3 \) using the power rule?

  3x^2
• What is the process of finding the derivative of a function called?

  Differentiation
• If the derivative of a function is positive, the function is increasing.


  True
• Arrange the derivatives of trigonometric functions:

  1️⃣ Derivative of \(\sin x\) is \(\cos x\)

  2️⃣ Derivative of \(\cos x\) is \(-\sin x\)

  3️⃣ Derivative of \(\tan x\) is \(\sec^2 x\)
• What is the derivative of \( \sin x \)?

  \(\cos x\)
• What is the derivative of \( \ln x \)?

  \(\frac{1}{x}\)
• Match the differentiation rule with its description:

  Power Rule ↔️ If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \)

  Sum Rule ↔️ If \( f(x) = u(x) + v(x) \), then \( f'(x) = u'(x) + v'(x) \)

  Constant Rule ↔️ If \( f(x) = c \), then \( f'(x) = 0 \)
• What is differentiation the process of finding?

  The derivative
• For \(f(x) = x^2\), the derivative is \(f'(x) = 2x\). At \(x = 2\), \(f'(2) = 4\), indicating the function's slope at \(x = 2\) is 4
• What does a derivative of zero indicate about the function's behavior?

  Local extremum
• What does a positive derivative signify about the function's behavior?

  Function is increasing
• The derivative of a constant function is always zero.

  True
• If \( f(x) = c \), where \( c \) is a constant, then \( f'(x) = \)0
• A polynomial function is expressed as the sum of terms with a coefficient and a variable raised to a non-negative integer power
• The derivative of a function represents the rate of change of the function.
• The derivative of a function can be interpreted as the slope of the tangent line to the function at a particular point.
• The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \), which simplifies finding the derivative of terms with powers.
• The derivative of a constant \( c \) is 0
• The derivative of \( a^x \) is \( a^x \ln(a) \)


  True
• The derivative of a function represents its rate of change
• The derivative represents the instantaneous rate of change of a function.

  True
• Match the derivative sign with the function's behavior:

  Positive ↔️ Increasing

  Negative ↔️ Decreasing

  Zero ↔️ Local extremum
• A positive derivative indicates that the function is increasing.


  True
• What is the derivative of \( f(x) = x^2 + 3x \) using the sum rule?

  2x + 3
• Match the differentiation rule with its description:

  Power Rule ↔️ \( f(x) = x^n \implies f'(x) = nx^{n-1} \)

  Sum Rule ↔️ \( f(x) = u(x) + v(x) \implies f'(x) = u'(x) + v'(x) \)

  Constant Rule ↔️ \( f(x) = c \implies f'(x) = 0 \)
• What is the derivative of \( f(x) = \sin x \)?

  \(\cos x\)
• What is the derivative of \( f(x) = a^x \)?

  \(a^x \ln(a)\)
• The derivative of \( f(x) = x^n \) is \( f'(x) = nx^{n-1} \).


  True
• What is the relationship between the slopes of a tangent and its normal at the same point?

  Perpendicular
• What is the formula to find the slope of the normal given the slope of the tangent?

  $m_{n} =$$- \frac{1}{m_{t}}$
• The slope of the normal is equal to the derivative of the function at the point of interest.

  False
• What is the equation of the tangent to \(f(x) = x^2\) at \(x = 2\)?

  y - 4 = 4(x - 2)</latex>
• The slope of the normal is the negative reciprocal of the slope of the tangent.
• What are critical points in optimization problems?

  Points where f'(x) = 0