Differentiation Rules: Fresh Take

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Differentiation Rules: Fresh Take

Use basic rules to differentiate functions that involve adding, subtracting, scaling, and raising to powers
Apply specific rules to find derivatives of functions multiplied or divided by each other
Use a combination of rules to calculate derivatives for polynomial and rational functions

The Basic Rules

The Main Idea

Constant Rule:
- For any constant $c$ , $\frac{d}{dx}(c) = 0$
Power Rule:
- For $f(x) = x^n$ where $n$ is a positive integer: $\frac{d}{dx}(x^n) = nx^{n-1}$
Sum and Difference Rules:
- $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$
- $\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)$
Constant Multiple Rule:
- For any constant $k$ , $\frac{d}{dx}(kf(x)) = kf'(x)$

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 $g(x)=-3$ .

$0$

Find $\frac{d}{dx}(x^4)$

Use $(x+h)^4=x^4+4x^3h+6x^2h^2+4xh^3+h^4$ .

$4x^3$

Find the derivative of $f(x)=x^7$ .

Use the power rule with $n=7$ .

$f^{\prime}(x)=7x^6$

Watch the following video to see the worked solution to this example.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “3.3 Differentiation Rules” here (opens in new window).

Find the derivative of $f(x)=2x^3-6x^2+3$ .

$f^{\prime}(x)=6x^2-12x$ .

Watch the following video to see the worked solution to this example.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “3.3 Differentiation Rules” here (opens in new window).

Find the equation of the line tangent to the graph of $f(x)=3x^2-11$ at $x=2$ . Use the point-slope form.

$y=12x-23$

The Advanced Rules

The Main Idea

Product Rule:
- For $j(x) = f(x)g(x)$ : $j'(x) = f'(x)g(x) + g'(x)f(x)$
Quotient Rule:
- For $j(x) = \frac{f(x)}{g(x)}$ : $j'(x) = \frac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^2}$
Extended Power Rule:
- For $k$ a negative integer: $\frac{d}{dx}(x^k) = kx^{k-1}$

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 $j(x)=2x^5(4x^2+x)$ .

Set $f(x)=2x^5$ and $g(x)=4x^2+x$ .

$j^{\prime}(x)=10x^4(4x^2+x)+(8x+1)(2x^5)=56x^6+12x^5$ .

Watch the following video to see the worked solution to this example.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “3.3 Differentiation Rules” here (opens in new window).

Find the derivative of $g(x)=\dfrac{1}{x^7}$ using the extended power rule.

Rewrite $g(x)=\frac{1}{x^7}=x^{-7}$ . Use the extended power rule with $k=-7$ .

$g^{\prime}(x)=-7x^{-8}$ .

Watch the following video to see the worked solution to this example.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “3.3 Differentiation Rules” here (opens in new window).

Find the derivative of $h(x) = (x^3 + 2x)(4x^2 - 3)$ .

Let $f(x) = x^3 + 2x$ and $g(x) = 4x^2 - 3$

$\begin{array}{rcl} f'(x) &=& 3x^2 + 2 \\ g'(x) &=& 8x \end{array}$

Applying the product rule:

$\begin{array}{rcl} h'(x) &=& f'(x)g(x) + g'(x)f(x) \\ &=& (3x^2 + 2)(4x^2 - 3) + (8x)(x^3 + 2x) \\ &=& (12x^4 - 9x^2 + 8x^2 - 6) + (8x^4 + 16x^2) \\ &=& 20x^4 + 15x^2 - 6 \end{array}$

Find the derivative of $k(x) = \frac{x^2 + 3x}{2x - 1}$ .

Let $f(x) = x^2 + 3x$ and $g(x) = 2x - 1$

$\begin{array}{rcl} f'(x) &=& 2x + 3 \\ g'(x) &=& 2 \end{array}$

Applying the quotient rule:

$\begin{array}{rcl} k'(x) &=& \dfrac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^2} \\ &=& \dfrac{(2x + 3)(2x - 1) - 2(x^2 + 3x)}{(2x - 1)^2} \\ &=& \dfrac{4x^2 - 2x + 6x - 3 - 2x^2 - 6x}{(2x - 1)^2} \\ &=& \dfrac{2x^2 - 2x - 3}{(2x - 1)^2} \end{array}$

Find the derivative of $f(x) = 5x^{-3} - 2x^{-1}$ .

Using the extended power rule and the sum rule:

$\begin{array}{rcl} f'(x) &=& \frac{d}{dx}(5x^{-3}) - \frac{d}{dx}(2x^{-1}) \\ &=& 5(-3x^{-4}) - 2(-1x^{-2}) \\ &=& -15x^{-4} + 2x^{-2} \end{array}$

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 $k(x)=f(x)g(x)h(x)$ , express $k^{\prime}(x)$ in terms of $f(x), \, g(x), \, h(x)$ , and their derivatives.

We can think of the function $k(x)$ as the product of the function $f(x)g(x)$ and the function $h(x)$ . That is, $k(x)=(f(x)g(x))\cdot h(x)$ . Thus,

$\begin{array}{lllll}k^{\prime}(x) & =\frac{d}{dx}(f(x)g(x))\cdot h(x)+\frac{d}{dx}(h(x))\cdot (f(x)g(x)) & & & \begin{array}{l}\text{Apply the product rule to the product} \\ \text{of} \, f(x)g(x) \, \text{and} \, h(x). \end{array} \\ & =(f^{\prime}(x)g(x)+g^{\prime}(x)f(x))h(x)+h^{\prime}(x)f(x)g(x) & & & \text{Apply the product rule to} \, f(x)g(x). \\ & =f^{\prime}(x)g(x)h(x)+f(x)g^{\prime}(x)h(x)+f(x)g(x)h^{\prime}(x). & & & \text{Simplify.} \end{array}$

Watch the following video to see the worked solution to this example.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “3.3 Differentiation Rules” here (opens in new window).

Find $\frac{d}{dx}(3f(x)-2g(x))$ .

$3f^{\prime}(x)-2g^{\prime}(x)$ .

Find the value(s) of $x$ for which the line tangent to the graph of $f(x)=4x^2-3x+2$ is parallel to the line $y=2x+3$ .

Solve the equation $f^{\prime}(x)=2$ .

$\frac{5}{8}$