1.7 Differentiation

• What is the formula for finding the derivative using the first principle?

  $f'(x) =$$\lim_{h\to0} \frac{f(x + h) - f(x)}{h}$
• The derivative of $x^{3}$ using the power rule is $3x^{2}$.


  True
• What is the exponent in the function `x^n` called?

  n
• Differentiation geometrically represents the gradient of a function at any given point.


  True
• There are three main notations used in differentiation: prime, Leibniz, and operator notation.

  True
• The operator notation for differentiation is written as `D[f](x)`
• When applying the power rule, you multiply the exponent `n` by the coefficient
• The derivative of \(\cos x\) is \(\sin x\).

  False
• The derivative of \(\ln x\) is \(\frac{1}{x}\)
• What is the derivative of $\ln x$?


  $1 / x$
• The derivative represents the rate of change of the function with respect to its variable
• The most common notations for the derivative are prime notation, Leibniz notation, and operator notation
• Match the notation with its meaning:

  $dy / dx$ ↔️ Rate of change of $y$ with respect to $x$

  $f'(x)$ ↔️ First derivative of $f$ with respect to $x$

  $D[f](x)$ ↔️ Derivative operator applied to $f$
• Steps to apply the power rule:

  1️⃣ Identify the exponent $n$ of the function x^n</latex>

  2️⃣ Multiply the exponent $n$ by the coefficient

  3️⃣ Reduce the exponent by 1 (from $n$ to $n - 1$)
• To apply the power rule, you reduce the exponent `n` to n-1
• The first principle formula for differentiation is `f'(x) = lim h→0 (f(x+h) - f(x)) / h
• What does the Leibniz notation `dy/dx` represent?

  Rate of change of y with respect to x
• The power rule states that the derivative of `x^n` is `nx^(n-1)`.


  True
• Match the trigonometric function with its derivative:

  \(\sin x\) ↔️ \(\cos x\)

  \(\cos x\) ↔️ \(-\sin x\)

  \(\tan x\) ↔️ \(\sec^2 x\)
• What is the derivative of $e^{x}$?


  $e^{x}$
• The derivative of $a^{x}$ is a^x \ln(a)</latex>
• The derivative of $\log_{a} x$ is 1 / (x \ln(a))</latex>.


  True
• What is the process of finding the rate of change of a function called?

  Differentiation
• When applying the power rule, you first identify the exponent of the function.

  True
• The gradient of the line $y =$$3x +$$2$ is 3.


  True
• What does the prime notation f'(x)</latex> indicate?

  The first derivative
• The power rule states that the derivative of $x^{n}$ is $nx^{n - 1}$
• The derivative of $x^{5}$ is $5x^{4}$
• What is the derivative of `x^5` using the power rule?

  $5x^{4}$
• What does the derivative of a function represent in terms of rate of change?

  Instantaneous rate of change
• What is the derivative of `x^5` using the power rule?

  $5x^{4}$
• What is the derivative of \(\sin x\)?

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

  $e^{x}$
• What does the derivative of a function represent geometrically?

  Gradient of the function
• Match the differentiation notation with its description:

  Prime Notation (`f'(x)`) ↔️ Indicates the first derivative of a function `f` with respect to `x`

  Leibniz Notation (`dy/dx`) ↔️ Shows the rate of change of `y` with respect to `x`

  Operator Notation (`D[f](x)`) ↔️ Uses `D` to denote the derivative operator applied to function `f`
• The power rule states that the derivative of x^n is
• If `f(x) = x^5`, what is its derivative using the power rule?

  5x^4
• The chain rule states that if \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\)


  True
• The derivative of a^x is $a^{x} \ln(a)$
• Steps to apply the power rule:

  1️⃣ Identify the exponent n of the function x^n

  2️⃣ Multiply the exponent n by the coefficient

  3️⃣ Reduce the exponent by 1