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

step 1 Okay, so for the first part, we're going to use the quotient rule. And in that, what you do is y

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

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

Key Concepts

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.

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.

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.

Transcript

00:01 Okay, so for the first part, we're going to use the quotient rule.

00:04 And in that, what you do is you take the derivative of the numerator, which in this case is 10x minus 1.

00:14 So the derivative of 10x is 2.

00:16 So you get squared x, and then derivative of 1 is 0.

00:21 And then you write the denominator as it is minus the numerator as it is and the derivative of the denominator denominator which is secant x 10 x and this entire thing gets divided by secant squared x and when you simplify and rewrite this final answer is secant x minus 10xx minus 1 into tan x over secant x so that's the first part now if we go to b part another way to solve this is to simplify the given function so this is the given function and we know that tan x is sine x over cosine x minus 1 over secant is 1 over cosine x.

01:41 Now if we separate the terms, you get sine x over, oops, cosine x over 1 over cosine x minus 1 over 1 over cosine x.

02:00 So here the cosines cancel.

02:04 So this is going to equal so sine x minus cosine x.

02:16 And if you take the derivative of that, the derivative of sine is cosine, and the derivative of cosine is minus sine.

02:29 So the final answer is cosine x plus sine x.

02:37 Now to show that part a and part b are actually the same answer, we can try and simplify the answer that we got in part a.

02:47 So let's rewrite what we got in part a, which is second x minus 10x minus 1 into tan x over secant x.

03:07 So let's try and simplify this.

03:12 Let's write the first term as it is...

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