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Solve the given problems.Find the slope of a line tangent to the curve of $y=\frac{2 \sin 3 x}{x},$ where $x=0.15 .$ Verify the result by using the derivative-evaluating feature of a calculator.

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 1

Derivatives of the sine and Cosine Functions

Derivatives

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03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

01:43

Compute the slope of the t…

01:10

Find the slope of the tang…

01:24

04:06

02:25

Find an equation for the t…

04:08

Differentiate the function…

02:17

Find the equation of the t…

00:24

Find an equation of the ta…

02:39

Find an equation of the li…

02:34

Find all values of $x$ at …

Okay. This question is asked us to find the slope of the tangent line with this equation at X equals 0.15 And then verify it using the derivative finding function on the calculated. Okay, so we need to take the derivative clearly, we're going to use quotation role. So why prime equals X. The denominator times the derivative of the top. The derivative of the top will be six. Co sign three X. That three popping out in front minus the derivative. The bottom which is one times the original top which is to sign three X. All over the denominator squared. Ok. We don't have to worry about simplifying this down too much because we're substituting a number in here. We're substituting in that 0.15 15 times three will be a 30.45 Make a little bit easier calculations. Okay, substituting in 0.15 Sorry about that everywhere we see an X. Yeah, we're dividing by 0.5 square. Okay, we use that calculator for this number crunching So set up a fraction 0.15 six co sign 0.45 Okay, my ass to sign 0.45 divided by 0.15 squared gives us negative 2646 Oh. This is the slope of the tangent line. We can just check that. Using the derivative. Finding function on a calculator. Okay, on the T I 84 I'll show you where that is. Second. Okay, we're gonna go into math, we're gonna go down here to end the riff. We'll find us a numerical derivative. We're looking for DDX we're gonna put our function in here. We'll set it up as a fraction is to sign three X over X. And we wanted the numerical derivative at 0.15 So we put that in here and there, it confirms it for us. So, slope of the tangent line at that point negative 2.646

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