Unsure on how to approach sin^4

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In summary, for d/dx (cos 4x + sin^(4) x), the derivative is -4sin 4x + 4sin^3 x cos x, using both the power rule and the chain rule. For d/dx (1)/(16-t^2)^(1/4), the derivative can be written as -1/4(16-t^2)^(-5/4)(-2t) or simply as (16-t^2)^(-1/4), using the chain rule directly.
  • #1
helpm3pl3ase
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For d/dx (cos 4x + sin^(4) x)

is it.

-sin(4x)(4) + cos^4 (x)? I was unsure on how to approach sin^4. Thanks.

Also for d/dx (1)/(16-t^2)^(1/4)

Is this correct: (-1/4 (16-t^2)^(-5/4))\((16-t^2)^(1/4))^2??

Thank you.
 
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  • #2
For (1) [tex] \frac{d}{dx} \cos 4x + \sin^{4} x = -4\sin 4x + 4\sin^{3} x \cos x [/tex]. You have to use both the power rule and the chain rule.

[tex] \frac{d}{dx} \sin^{4}x, \ u = \sin x [/tex]

[tex] \frac{d}{du} u^{4} du = 4u^{3} du [/tex]For (2) the first half is right. The second half, you have to use the chain rule.
 
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  • #3
Y use the chain rule isn't the second half the bottom of the quotient rule, so it stays like that?/
 
  • #4
[tex] \frac{d}{dt} (16-t^{2})^{-\frac{1}{4}} = -\frac{1}{4}(16-t^{2})^{-\frac{5}{4}}(-2t) [/tex]. So we have used the chain rule.
 
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  • #5
O that goes on the top.. the bottom is correct tho right?
 
  • #6
quote= helpm3pl3ase]Also for d/dx (1)/(16-t^2)^(1/4)

Is this correct: (-1/4 (16-t^2)^(-5/4))\((16-t^2)^(1/4))^2??[/quote]
You can use the quotient rule (with the chain rule):
[tex]\frac{(d/dx(1))(16-t^2)^{1/4}- (1)(d/dx(16-t^2)^{1/4})}{(16-t^2)^{1/2}}[/tex]
[tex]= -\frac{(1)(1/4)(16-t^2)^{-3/4}(-2t)}{(16-t^2)^{1/2}}[/tex]
[tex]= \frac{1}{2}\frac{t}{(16-t^2)^{5/4}}[/tex]

But it is much easier to write the function as
[tex](16- t^2)^{-1/4}[/itex]
and use the chain rule directly:
[tex]d/dx(16- t^2)^{-1/4}= -1/4(16- t^2)^{-5/4}(-2t)[/tex]
[tex]= \frac{1}{2}\frac{t}{(16-t^2)^{5/4}}[/tex]
 

Related to Unsure on how to approach sin^4

1. What is sin^4 and how is it different from regular sin?

Sin^4 is a mathematical function that represents taking the fourth power of the sine of an angle. This is different from regular sin, which only represents taking the first power or the sine of an angle.

2. How should I approach solving a problem involving sin^4?

When approaching a problem involving sin^4, it is important to remember to first take the fourth power of the sine of the given angle. Then, solve the rest of the equation as you would with any other mathematical problem.

3. Can I use a calculator to solve problems involving sin^4?

Yes, most scientific calculators have a button or function specifically for calculating the fourth power of a number. Just make sure to input the angle in radians if necessary.

4. What are some common applications of sin^4 in science?

Sin^4 is commonly used in physics, specifically in calculating the energy of a wave. It is also used in electrical engineering to calculate the power output of a circuit.

5. Are there any other trigonometric functions that can be raised to the fourth power?

Yes, other trigonometric functions such as cosine and tangent can also be raised to the fourth power. However, these are not as commonly used as sin^4 in scientific applications.

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