#Derivatives and Tangents

#Table of Contents

1. What is the Derivative of a Function?
2. How are Derivatives and Tangents Related?
3. Finding the Equation of a Tangent to a Curve
4. Horizontal and Vertical Tangents
5. Worked Example
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

#What is the Derivative of a Function?

The derivative of a function describes the instantaneous rate of change of the function at any given point. It is equal to the slope of the curve at that point.

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

This definition is valid for values of $x$ where this limit exists. Note that the derivative is also a function of $x$.

There are several ways to denote the derivative of $f(x)$:

• $f'(x)$
• $\frac{dy}{dx}$
• $y'$

You may also see "the derivative of ..." written as " ... differentiated". They mean the same thing.

#How are Derivatives and Tangents Related?

The value of the derivative of a function at a point is equal to the slope of the tangent to the graph at that point.

#Finding the Equation of a Tangent to a Curve

To find the equation of a tangent line to the graph of a function $f$ at the point $(a, b)$ using a derivative:

1. Represent the equation of the tangent using the general form for the equation of a straight line with slope $m$ that goes through point $(x_1, y_1)$: $$y - y_1 = m(x - x_1)$$
2. Substitute in $(x_1, y_1)$. This is the point the tangent touches on the curve.
3. Find the value of the derivative of $f(x)$ at the point $(a, b)$ if it is not given; this is $f'(a)$. The value of the derivative of $f(x)$ at $(a, b)$ is equal to the slope of the tangent at $(a, b)$.
4. Substitute $m$ in the equation of the tangent: $$y - b = f'(a)(x - a)$$
5. This equation can then be rearranged to another desired form if needed, for example: $$y = b + f'(a)(x - a)$$

#Horizontal and Vertical Tangents

The tangent line to the graph of a function $f$ will be horizontal when $f'(x) = 0$. Horizontal lines have a slope of zero.

The tangent line to the graph of a function $f$ will be vertical when $f'(x)$ is undefined because of dividing a constant by zero. For example, for $f(x) = x^{1/3}$ with derivative $f'(x) = \frac{1}{3x^{2/3}}$:

• $f(x)$ is defined at $x = 0$.
• Therefore, the graph of $f$ has a vertical tangent at $x = 0$.

Don't confuse vertical tangents with vertical asymptotes. Tangents and curves intersect, but curves only approach asymptotes without ever intersecting with them.

#Worked Example

Let the function $f$ be defined by $f(x) = x^2 - 3x - 4$.

Find the equation of the line that is tangent to the graph of $f$ at the point where $x = 4$.

Answer:

The tangent is a straight line of the form $y - y_1 = m(x - x_1)$.

The question states that the instantaneous rate of change (the slope) of the curve when $x = 4$ is 5: $$f'(4) = 5$$

This means the slope of the tangent, $m$, will also be 5 at this point. The $x$-coordinate of the point is known, but not the $y$ value: $$(4, y_1)$$

Find $f(x)$: $$y_1 = f(4) = 4^2 - 3(4) - 4 = 4$$

So the point where the tangent touches the curve is $(4, 4)$. Substitute the point, and the slope at this point, into the equation of the tangent: $$y - 4 = 5(x - 4)$$

Simplify: $$y = 5(x - 4) + 4$$

Or: $$y = 5x - 20 + 4 \implies y = 5x - 16$$

#Practice Questions

Practice Question

1. Find the derivative of the function $f(x) = 2x^3 - 3x^2 + 4x - 5$.

2. Determine the slope of the tangent line to the curve $f(x) = x^2 - 4x + 7$ at the point where $x = 2$.

3. Find the equation of the tangent line to the graph of $f(x) = \sqrt{x}$ at the point $(1, 1)$.

4. Determine whether the tangent line to the curve $f(x) = \frac{1}{x}$ at $x = 1$ is horizontal, vertical, or neither.

5. Calculate the derivative of $f(x) = \ln(x)$ and find the slope of the tangent line at $x = e$.

#Glossary

• Derivative: The measure of how a function changes as its input changes, represented as $f'(x)$ or $\frac{dy}{dx}$.
• Tangent: A line that touches a curve at a single point without crossing it at that point.
• Slope: The measure of the steepness of a line, calculated as the ratio of the vertical change to the horizontal change.
• Instantaneous Rate of Change: The rate of change of a function at a specific point, equivalent to the derivative at that point.
• Horizontal Tangent: A tangent line that has a slope of zero.
• Vertical Tangent: A tangent line that is vertical, occurring where the derivative is undefined.

#Summary and Key Takeaways

• The derivative of a function gives the instantaneous rate of change and is equal to the slope of the tangent to the curve at that point.
• The equation of a tangent line can be found using the derivative and the point of tangency.
• Tangent lines can be horizontal (slope = 0) or vertical (undefined slope).
• Understanding derivatives and tangents is crucial for solving many problems in calculus, including finding rates of change and optimizing functions.