Trig Function With an Exponent

[Jason76]

Trig Function with an Exponent

1.

\(\displaystyle y = \sec^{4} 3x\)

\(\displaystyle y' = 4 \sec^{3} 3x(\sec 3x \tan 3x)(3)dx\) This line makes sense. But how does it get to the next line?

\(\displaystyle y' = 12 \sec^{4} 3x \tan 3x dx\)

Shouldn't it be:

\(\displaystyle y' = 12 \sec^{3} 3x(\sec 3x \tan 3x) dx\) ?

This other example is different and I totally understand it:

2.

\(\displaystyle y = \cot^{4} 2x\)

\(\displaystyle y' = 4(\cot^{3} 2x)(-\csc^{2} 2x)(2) dx\)

\(\displaystyle y' = -8(\cot^{3} 2x)(\csc^{2} 2x) dx\)

What's the difference between these two examples?

 

[Deleted member 4993]

1.

\(\displaystyle y = \sec^{4} 3x\)

\(\displaystyle y' = 4 \sec^{3} 3x(\sec 3x \tan 3x)(3)dx\) This line makes sense. But how does it get to the next line?

Simple algebra!!

a³ * a = a³ * a¹ = a⁽³⁺¹⁾ = a⁴

Same way:

sec³(3x) * sec(3x) = sec⁴(3x)


\(\displaystyle y' = 12 \sec^{4} 3x \tan 3x dx\)

Shouldn't it be:

\(\displaystyle y' = 12 \sec^{3} 3x(\sec 3x \tan 3x) dx\) ?

This other example is different and I totally understand it:

2.

\(\displaystyle y = \cot^{4} 2x\)

\(\displaystyle y' = 4(\cot^{3} 2x)(-\csc^{2} 2x)(2) dx\)

\(\displaystyle y' = -8(\cot^{3} 2x)(\csc^{2} 2x) dx\)

What's the difference between these two examples?.................. you tell us !!

.

 

[Jason76]

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If the distributive property is going on, then shouldn't the 3x terms be multiplied and become 9x terms? Never mind, no plus sign so no distributive property. Why aren't the 3x terms multiplied? I can see the exponent thing now with the trig functions.

 

[Deleted member 4993]

If the distributive property is going on, then shouldn't the 3x terms be multiplied and become 9x terms? Never mind, no plus sign so no distributive property. Why aren't the 3x terms multiplied? I can see the exponent thing now with the trig functions.

You need to review your knowledge of functions.

First you need to recognize f²(x) is, in general, different from f(x²)

Then you need to recognize:

f(x) * f(x) = f²(x) and, in general, \(\displaystyle \ne\) f(x²) and, in general,\(\displaystyle \ne\) f²(x²)