I understand that the derivative of cosx is -sinx, but I need to show the work using trig identities. I worked out a bit of it, but i can't figure out the rest.
I understand that the derivative of cosx is -sinx, but I need to show the work using trig identities. I worked out a bit of it, but i can't figure out the rest.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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I understand that the derivative of cosx is -sinx, but I need to show the work using trig identities. I worked out a bit of it, but i can't figure out the rest.
![**Derivative of Cosine Function**
To demonstrate that the derivative of \( \cos(x) \) is \( -\sin(x) \), we use the limit definition of the derivative.
1. Start with the definition:
\[
\frac{d}{dx} \cos(x) = \lim_{h \to 0} \frac{\cos(x+h) - \cos(x)}{h}
\]
2. Apply the cosine addition formula:
\[
\cos(x+h) = \cos(x)\cos(h) - \sin(x)\sin(h)
\]
3. Substitute into the limit:
\[
\lim_{h \to 0} \frac{\cos(x)\cos(h) - \sin(x)\sin(h) - \cos(x)}{h}
\]
4. Rearrange terms:
\[
\lim_{h \to 0} \frac{\cos(x)(\cos(h) - 1) - \sin(x)\sin(h)}{h}
\]
5. Split the limit:
\[
\lim_{h \to 0} \left[ \frac{\cos(x)(\cos(h) - 1)}{h} - \frac{\sin(x)\sin(h)}{h} \right]
\]
6. Evaluate each term:
- The first term: \( \lim_{h \to 0} \frac{\cos(h) - 1}{h} = 0 \)
- The second term: \( \lim_{h \to 0} \frac{\sin(h)}{h} = 1 \)
7. Substitute these evaluations back:
\[
\cos(x) \cdot 0 - \sin(x) \cdot 1 = -\sin(x)
\]
Therefore, the derivative of \( \cos(x) \) is indeed \( -\sin(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7efc92fb-f6e6-45fd-bcd8-2c3c0a4ba455%2F0537aea6-76e1-4d4a-90b8-079639ca1c4b%2F6gti40f.jpeg&w=3840&q=75)
Transcribed Image Text:**Derivative of Cosine Function**
To demonstrate that the derivative of \( \cos(x) \) is \( -\sin(x) \), we use the limit definition of the derivative.
1. Start with the definition:
\[
\frac{d}{dx} \cos(x) = \lim_{h \to 0} \frac{\cos(x+h) - \cos(x)}{h}
\]
2. Apply the cosine addition formula:
\[
\cos(x+h) = \cos(x)\cos(h) - \sin(x)\sin(h)
\]
3. Substitute into the limit:
\[
\lim_{h \to 0} \frac{\cos(x)\cos(h) - \sin(x)\sin(h) - \cos(x)}{h}
\]
4. Rearrange terms:
\[
\lim_{h \to 0} \frac{\cos(x)(\cos(h) - 1) - \sin(x)\sin(h)}{h}
\]
5. Split the limit:
\[
\lim_{h \to 0} \left[ \frac{\cos(x)(\cos(h) - 1)}{h} - \frac{\sin(x)\sin(h)}{h} \right]
\]
6. Evaluate each term:
- The first term: \( \lim_{h \to 0} \frac{\cos(h) - 1}{h} = 0 \)
- The second term: \( \lim_{h \to 0} \frac{\sin(h)}{h} = 1 \)
7. Substitute these evaluations back:
\[
\cos(x) \cdot 0 - \sin(x) \cdot 1 = -\sin(x)
\]
Therefore, the derivative of \( \cos(x) \) is indeed \( -\sin(x) \).
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