(a) Use the Quotient Rule to differentiate the function f(x)=(tanx-1)/(secx) (b) Simplify t

(a) Use the Quotient Rule to differentiate the function...

(a) Use the Quotient Rule to differentiate the function $$f(x)=\frac{\tan x-1}{\sec x}$$ $\begin{array}{l}{\text { (b) Simplify the expression for } f(x) \text { by writing it in terms }} \\ {\text { of sin } x \text { and } \cos x, \text { and then find } f^{\prime}(x) \text { . }} \\ {\text { (c) Show that your answers to parts (a) and (b) are }} \\ {\text { equivalent. }}\end{array} $

(a) Use the Quotient Rule to differentiate the function
$$f(x)=\frac{\tan x-1}{\sec x}$$
$\begin{array}{l}{\text { (b) Simplify the expression for } f(x) \text { by writing it in terms }} \\ {\text { of sin } x \text { and } \cos x, \text { and then find } f^{\prime}(x) \text { . }} \\ {\text { (c) Show that your answers to parts (a) and (b) are }} \\ {\text { equivalent. }}\end{array}
$

Key Concepts

Equivalence Verification

A process in calculus where two different methods or forms of deriving a result are compared to ensure they yield the same outcome. This concept highlights the consistency of mathematical operations and validates that alternative approaches to differentiation are correctly applied.

Trigonometric Identities

These are universal relationships between trigonometric functions that help in rewriting and simplifying expressions. For instance, converting secant into cosine or expressing tangent in terms of sine and cosine allows for easier manipulation and differentiation.

Expression Simplification

This involves using algebraic techniques and identities to rewrite expressions in a more manageable or revealing form. In the context of trigonometric functions, simplifying expressions often leads to more straightforward differentiation and clearer insights into the function's behavior.

Quotient Rule

A technique in differential calculus used to differentiate functions that are expressed as the ratio of two differentiable functions. It provides a formula to compute the derivative by taking into account both the derivative of the numerator and the denominator.

Differentiation of Trigonometric Functions

This concept involves applying standard differentiation rules to trigonometric functions such as sin, cos, tan, and sec. Understanding these derivatives is fundamental for handling more complex expressions that involve trigonometric components.

Transcript

00:02 Alright, in this problem, i'm going to take the derivative of this using the quotient rule.

00:07 Then i'm going to switch it to signs and cosines and take the derivative again, and then i'm going to show you that the two derivatives are the same.

00:19 Okay, this is a quotient, so i have to use the question rule.

00:22 It's the bottom times the derivative of the top, and the derivative of the tangent of x is secan squared of x, minus the derivative of 1, which is 0.

00:34 So bottom, derivative of the top, minus the top, times the derivative of the bottom and the derivative of secan of x is secan x tangent x and then the whole thing over the bottom squared so i'm going to go ahead and multiply the top out so i have secant to the third x there i'm going to distribute i get minus sikin x tangent squared x minus minus which makes plus secan x tangent x all over sikin squared x.

01:23 Now i notice that every piece on the top has a sikin in it, so i'm going to factor out the sikin of x.

01:31 So i get sikin squared x minus tangent squared x plus tangent x, all over sikin squared x.

01:48 All right, we know the identity 1 plus tangent squared x equals siken squared x.

01:56 So then sqn squared minus tangent squared equals to 1.

02:11 So one of these sequins will cancel with one of these, or this sequence will cancel with one of those.

02:18 So now we have one plus tangent x over sicken x.

02:26 Okay, so that's the derivative, the first way.

02:35 Now i'm going to take the problem, f of x equals tangent x minus one over sicken x.

02:45 I'm going to change everything to sine and cosine.

02:47 So i get sine x over cosine x minus 1 over 1 over the cosine of x.

02:58 So here i have a complex fraction or a four -story fraction.

03:02 And so what i'm going to do is simplify it by getting a common denominator for all the fractions and multiplying...

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