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Get Started Now- Intro Lesson: a7:08
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The derivative of a function is interpreted as the slope of the tangent line to the curve of the function at a certain given point. In this section, we will explore the meaning of a derivative of a function, as well as learning how to find the slope-point form of the equation of a tangent line, as well as normal lines, to a curve at multiple given points.

• *Point-Slope Form * of a line with slope m through a point $(x_1,y_1): m=\frac{y-y_1}{x-x_1}$

•__Tangent Line & Normal Line__

The**normal line** to a curve at a particular point is the line through that point and *perpendicular* to the **tangent line**.

•

The

- Introduction
__Connecting: Derivative & Slope & Equation of Tangent Line__

*Exercise:*The graph of the quadratic function $f\left( x \right) = \frac{1}{2}{x^2} + 2x - 1$ is shown below.

a)Find and interpret $f'\left( x \right)$.b)Find the slope of the tangent line at:

i) $x = - 1$

ii) $x = 2$

iii) $x = - 7$

iv) $x = - 4$

v) $x = - 2$c)Find an equation of the tangent line at:

i) $x = 2$

ii) $x = - 4$

iii) $x = - 2$

- 1.
**Determining Equations of the Tangent Line and Normal Line**

Consider the function: $f(x)=\frac{x}{32}(\sqrt{x}+{^3}\sqrt{x})$a)Determine an equation of the tangent line to the curve at $x=64$.b)Determine an equation of the normal line to the curve at $x=64$. - 2.
**Locating Horizontal Tangent Lines**a)Find the points on the graph of $f(x)=2x^3-3x^2-12x+8$ where the tangent is horizontal.b)Find the vertex of each quadratic function:

$f(x)=2x^2-12x+10$

$g(x)=-3x^2-60x-50$ - 3.
**Locating Tangent Lines Parallel to a Linear Function**

Consider the Cubic function: $f(x)=x^3-3x^2+3x$

i) Find the points on the curve where the tangent lines are parallel to the line $12x-y-9=0$.

ii) Determine the equations of these tangent lines. - 4.
**Determining Lines Passing Through a Point and Tangent to a Function**

Consider the quadratic function: $f(x)=x^2-x-2$a)Draw two lines through the point (3, -5) that are tangent to the parabola.b)Find the points where these tangent lines intersect the parabola.c)Determine the equations of both tangent lines. - 5.
**Locating Lines Simultaneously Tangent to 2 Curves**

Consider the quadratic functions:

$f(x)=x^2$

$g(x)=\frac{1}{4}x^2+3$a)Sketch each parabola.b)Determine the lines that are tangent to both curves.

29.

Derivatives

29.1

Definition of derivative

29.2

Power rule

29.3

Gradient and equation of tangent line

29.4

Chain rule

29.5

Derivative of trigonometric functions

29.6

Derivative of exponential functions

29.7

Product rule

29.8

Quotient rule

29.9

Implicit differentiation

29.10

Derivative of inverse trigonometric functions

29.11

Derivative of logarithmic functions

29.12

Higher order derivatives

We have over 1040 practice questions in AU Year 12 Maths for you to master.

Get Started Now29.1

Definition of derivative

29.2

Power rule

29.3

Gradient and equation of tangent line

29.4

Chain rule

29.5

Derivative of trigonometric functions

29.6

Derivative of exponential functions

29.7

Product rule

29.8

Quotient rule

29.9

Implicit differentiation

29.10

Derivative of inverse trigonometric functions

29.11

Derivative of logarithmic functions

29.12

Higher order derivatives