4.1 Differentiation

• The definition of differentiation calculates the instantaneous rate of change of a function.
• The formula for the definition of differentiation is $\frac{dy}{dx} =$$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.
• What does the definition of differentiation find as $h$ approaches zero?


  The derivative
• The derivative of $y =$$x^{2}$ using the definition of differentiation is 2x.
• The derivative of $y =$$3x +$$2$ using the definition of differentiation is 3.
• Match the function with its derivative using the definition of differentiation:

  x^{2}</latex> ↔️ $2x$

  $3x +$$2$ ↔️ $3$
• The power rule states that $\frac{d}{dx}(x^{n}) =$$nx^{n - 1}$, so the derivative of $3x^{4}$ is 12x^{3}.
• The derivative of $\sin x$ is $\cos x$, and the derivative of $\cos x$ is $- \sin x$.
• What is the derivative of $e^{x}$?


  $e^{x}$
• The derivative of $\ln x$ is $\frac{1}{x}$.
• Match the function with its differentiation rule and its derivative:

  $x^{n}$ ↔️ $\frac{d}{dx}(x^{n}) =$$nx^{n - 1}$

  $\sin x$ ↔️ $\frac{d}{dx}(\sin x) =$$\cos x$

  $e^{x}$ ↔️ $\frac{d}{dx}(e^{x}) =$$e^{x}$

  $\ln x$ ↔️ $\frac{d}{dx}(\ln x) =$$\frac{1}{x}$
• The product rule states that $\frac{d}{dx}(uv) =$$u'v +$$uv'$, where $u$ and $v$ are differentiable functions.
• The derivative of $x^{2} \sin x$ using the product rule is $2x \sin x +$$x^{2} \cos x$.
• What are u</latex> and $v$ in the product rule for differentiating $e^{x} \ln x$?


  $u =$$e^{x}$, $v =$$\ln x$
• Steps to apply the product rule for $y =$$e^{x} \ln x$


  1️⃣ Identify $u =$$e^{x}$ and $v =$$\ln x$

  2️⃣ Differentiate $u' =$$e^{x}$ and $v' =$$\frac{1}{x}$

  3️⃣ Apply the product rule: $\frac{dy}{dx} =$$u'v +$$uv'$

  4️⃣ Simplify: $\frac{dy}{dx} =$$e^{x} \ln x +$$\frac{e^{x}}{x}$
• The derivative of $x^{2} \sin x$ is 2x \sin x + x^{2} \cos x</latex>.
• What does the formula $\frac{dy}{dx} =$$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ calculate?


  Derivative of f(x)
• \frac{dy}{dx} = \lim_{h \to 0} (2x + h) = 2x
• What is the derivative of $3x +$$2$?


  3
• Match the function with its derivative:

  $x^{2}$ ↔️ $2x$

  $3x +$$2$ ↔️ $3$
• The definition of differentiation calculates the instantaneous rate of change of a function.
• The derivative of $y =$$f(x)$ is given by $\frac{dy}{dx} =$$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ and is also known as the limit of the difference quotient
• Order the common functions from the simplest to the most complex differentiation rule:

  1️⃣ Polynomials

  2️⃣ Trigonometric Functions

  3️⃣ Exponential Functions

  4️⃣ Logarithmic Functions
• What is the power rule for differentiating polynomials?

  $\frac{d}{dx}(x^{n}) =$$nx^{n - 1}$
• The derivative of \sin x</latex> is \cos x
• What is the derivative of $\ln x$?


  $\frac{1}{x}$
• Differentiation rules provide shortcuts for finding derivatives of common functions.
• Match the function with its derivative:

  $x^{n}$ ↔️ $nx^{n - 1}$

  $\sin x$ ↔️ $\cos x$

  $e^{x}$ ↔️ $e^{x}$

  $\ln x$ ↔️ $\frac{1}{x}$
• What is the product rule for differentiation?

  $\frac{d}{dx}(uv) =$$u'v +$$uv'$
• If $y =$$x^{2} \sin x$, then $u =$$x^{2}$ and $v =$$\sin x$, and their derivatives are $u' =$$2x$ and $v' =$$\cos x$. Applying the product rule, the derivative is 2x \sin x + x^{2} \cos x
• What is the derivative of $\ln x$?


  $\frac{1}{x}$
• The product rule states that the derivative of $uv$ is $u'v +$$uv'$.
• What is the product rule for differentiation?

  $\frac{d}{dx}(uv) =$$u'v +$$uv'$
• What is the derivative of $u =$$e^{x}$?


  $u' =$$e^{x}$
• The derivative of $v =$$\ln x$ is \frac{1}{x}</latex>
• The product rule states $\frac{d}{dx}(uv) =$$u'v +$$uv'$
• What is the derivative of $y =$$e^{x} \ln x$?


  $e^{x} \ln x +$$\frac{e^{x}}{x}$
• The derivative of $\sin x$ is $\cos x$
• What is the derivative of $y =$$x^{2} \sin x$?


  $2x \sin x +$$x^{2} \cos x$
• The quotient rule is used to differentiate the division of two differentiable functions.