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(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'(x). $(c) Show that your answers to parts (a) and (b) are equivalent.

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a) $f^{\prime}(x)=\frac{1+\tan x}{\sec x}$b) $f^{\prime}(x)=\cos x+\sin x$C) The part $(\mathrm{a})$ answer is equivalent to the part $(\mathrm{b})$ answer.

01:08

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 3

Derivatives of Trigonometric Functions

Derivatives

Differentiation

Campbell University

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

04:50

(a) Use the Quotient Rule …

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it's clear. So when you read here, so we're gonna use the quotient role we get seek it turns the derivative of tangent minus one minus tangent, minus one times the derivative of Seek it all over secrets where And then we get this to the equal to seek in times sequence square plus zero minus 10 times. Seek it. Times 10 all over. Seek and square Using our trig identities. We're gonna expand this so we get a second, learns vengeance Square, plus seek it minus seek it. Times turn gents Square was seeking time. 10 gin all over Secret Square in this equals one plus two engines over sequined per p. We have Earth of X is equal to 10 gin minus one divided by seeking. So using our trig identities, this is equal to sign over. Co sign minus one all over one over coastline. We're gonna multiply out the dominators. We got signed minus coastline. Next, we're gonna differentiate in respect to X, so we're gonna differentiate these separately. We get derivative of Sinus co sign the derivative of who signed is negative sign. We get co sign plus sign report. See, we're gonna do a we're gonna take our answer, which is one plus tangent over Seek in our first derivative. And we're gonna multiply by co sign for the top and bottom. And when we simplify, we get co sign plus sign. So we proved it.

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