- Find the derivative of a function.
- Find instantaneous rates of change.
- Find an equation of the tangent line to the graph of a function at a point.
Derivative Functions
The Main Idea
- Derivative Function:
- Gives the derivative of a function at every point in its domain
- Defined as: [latex]f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}[/latex]
- Differentiability:
- A function is differentiable at a point if its derivative exists at that point
- A function is differentiable on an interval if it’s differentiable at every point in that interval
- Notation for Derivatives:
- [latex]f'(x)[/latex], [latex]y'[/latex], [latex]\frac{dy}{dx}[/latex], [latex]\frac{d}{dx}(f(x))[/latex]
- Leibniz Notation:
- [latex]\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}[/latex]
- Represents instantaneous rate of change
Find the derivative of [latex]f(x)=x^2[/latex].
Watch the clip from 5:49 until 6:48. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.
You can view the transcript for this segmented clip of “3.2 The Derivative as a Function” here (opens in new window). If you would like to watch the entire video, you can view the transcript for “3.2 The Derivative as a Function” here (opens in new window).
You can view the transcript for “Derivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy” here (opens in new window).
The Basic Rules
The Main Idea
- Constant Rule:
- For any constant [latex]c[/latex], [latex]\frac{d}{dx}(c) = 0[/latex]
- Power Rule:
- For [latex]f(x) = x^n[/latex] where [latex]n[/latex] is a positive integer: [latex]\frac{d}{dx}(x^n) = nx^{n-1}[/latex]
- Sum and Difference Rules:
- [latex]\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)[/latex]
- [latex]\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)[/latex]
- Constant Multiple Rule:
- For any constant [latex]k[/latex], [latex]\frac{d}{dx}(kf(x)) = kf'(x)[/latex]
Key Concepts
- These rules form the foundation for differentiating more complex functions.
- The Power Rule applies to positive integer exponents and will be extended to other exponents later.
- These rules allow us to differentiate polynomials and many other functions without using the limit definition every time.
Find the derivative of [latex]g(x)=-3[/latex].
Find [latex]\frac{d}{dx}(x^4)[/latex]
Find the derivative of [latex]f(x)=x^7[/latex].
Find the derivative of [latex]f(x)=2x^3-6x^2+3[/latex].
Find the equation of the line tangent to the graph of [latex]f(x)=3x^2-11[/latex] at [latex]x=2[/latex]. Use the point-slope form.
The Advanced Rules
The Main Idea
- Product Rule:
- For [latex]j(x) = f(x)g(x)[/latex]: [latex]j'(x) = f'(x)g(x) + g'(x)f(x)[/latex]
- Quotient Rule:
- For [latex]j(x) = \frac{f(x)}{g(x)}[/latex]: [latex]j'(x) = \frac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^2}[/latex]
- Extended Power Rule:
- For [latex]k[/latex] a negative integer: [latex]\frac{d}{dx}(x^k) = kx^{k-1}[/latex]
Key Concepts
- The product rule is not simply the product of the derivatives.
- The quotient rule involves a specific arrangement of terms in the numerator.
- The extended power rule allows differentiation of negative integer powers.
- These rules expand our ability to differentiate more complex functions.
Use the product rule to obtain the derivative of [latex]j(x)=2x^5(4x^2+x)[/latex].
Find the derivative of [latex]g(x)=\dfrac{1}{x^7}[/latex] using the extended power rule.
Find the derivative of [latex]h(x) = (x^3 + 2x)(4x^2 - 3)[/latex].
Find the derivative of [latex]k(x) = \frac{x^2 + 3x}{2x - 1}[/latex].
Find the derivative of [latex]f(x) = 5x^{-3} - 2x^{-1}[/latex].
Combining Differentiation Rules
The Main Idea
- Combining Multiple Rules:
- Most real-world problems require applying several differentiation rules in sequence
- Apply rules in reverse order of function evaluation
- Problem-Solving Strategy:
- Identify the structure of the function
- Determine which rules apply and in what order
- Apply the rules systematically
- Simplify the result
- The order of applying differentiation rules matters
- Complex functions often require a combination of product, quotient, and basic rules
For [latex]k(x)=f(x)g(x)h(x)[/latex], express [latex]k^{\prime}(x)[/latex] in terms of [latex]f(x), \, g(x), \, h(x)[/latex], and their derivatives.
Find [latex]\frac{d}{dx}(3f(x)-2g(x))[/latex].
Find the value(s) of [latex]x[/latex] for which the line tangent to the graph of [latex]f(x)=4x^2-3x+2[/latex] is parallel to the line [latex]y=2x+3[/latex].