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

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

(a) Use the Ouotient 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

The quotient rule is a fundamental differentiation technique used when a function is expressed as the ratio of two differentiable functions. It states that if a function is given by f(x) = g(x)/h(x), then its derivative can be found using the formula f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]². This rule is particularly important for functions where direct simplification before differentiating is not immediately apparent.

Differentiation of Trigonometric Functions

Differentiating trigonometric functions involves recognizing the standard derivatives of functions like sine, cosine, tangent, and secant. Knowing these derivatives along with the application of the chain rule is essential when working with trigonometric function compositions or combinations, as these functions frequently appear in various calculus problems.

Simplification of Trigonometric Expressions

Expressing trigonometric functions in terms of sine and cosine can often simplify the process of differentiation. This technique allows for algebraic manipulation that might reduce a complex expression to a more manageable form, making it easier to apply differentiation rules and ultimately verifying the consistency between different methods of finding the derivative.

Equivalence of Differentiation Methods

Verifying that different differentiation approaches yield the same result is a common practice in calculus. Whether using the quotient rule directly or simplifying the function first before differentiating, this comparison ensures the correctness of the computation and deepens the understanding of the underlying relationships between various calculus techniques.

Transcript

00:01 All right, so we are deriving the following function, f of x equals tangent of x minus one divided by sicken of x.

00:06 Now for part a, they just want us to do it straight up.

00:09 So let's go ahead and do it.

00:12 Obviously, we need the quotient rule.

00:13 We are dividing two things.

00:15 So in order to do this, we do low, secant of x times d high, while the derivative of tangent of x minus 1 is secant squared of x.

00:26 So low d high, minus high, tangent of x minus 1 in parentheses, times d low, the derivative of seekin of x is seekin of x tangent of x.

00:44 And we divide that by the square of what's below, which is sequent squared of x.

00:52 Now, this thing is nasty, and you can kind of expect that because we didn't do the algebra up front, which we'll do in part b.

00:58 So let's try to simplify this.

01:01 Equals seagant cubed of x when we multiply that out and then what we can do up here is we can distribute this in and kind of hope for the best and also you know take care of the negative sign so we'll end up getting minus sequin of x tangent squared of x plus secan of x tangent of x that comes from distributing it, all divided by secant squared of x.

01:39 Now what we can do from here is cancel out at least one secant.

01:43 So what i'm going to do is split up the fraction on these signs right here, and i'm also going to cancel as many sequence as i can.

01:54 So just to spell that out for us, secant cubed over secant squared of x minus secant of x tangent squared of x over seekin squared of x plus secant of x tangent of x over secant squared of x.

02:20 Now i'm going to cancel as many sequins as i can, kind of like what i said.

02:27 This gives me seken of x right here minus tangent squared of x over secant of x plus tangent of x over seekin of x.

02:51 Now, something i'm going to do right here is write secant as one over cosine.

02:57 And that will kind of help us out in order to see what we're left with.

03:02 Sequent is one over cosine minus.

03:05 I'm even going to write tangent in terms of sine over cosine...

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