Derivative of Composite Trigonometric Functions

[kxpatel29]

Homework Statement

What is the derivative of f(x) = cos^5(sin(8x))

Homework Equations

trig, product rule, chain rule

The Attempt at a Solution

f(x) = cos^5(sin(8x))

Answer: 5-sin^4(cos(8x))(8)

 

[Mark44]

Homework Statement

What is the derivative of f(x) = cos^5(sin(8x))

Homework Equations

trig, product rule, chain rule

The Attempt at a Solution

f(x) = cos^5(sin(8x))

Answer: 5-sin^4(cos(8x))(8)

You're close. f'(x) = 5cos⁴(sin(8x))*cos(8x)*8
This is the same as 40cos(8x)sin⁴(sin(8x))

 

[spamiam]

Wait, isn't it

$$f'(x) = -40\cos^4(\sin(8x)) \cdot \sin(\sin(8x)) \cdot \cos(8x)$$
?

Because

$$\frac{d}{dx} (\cos(\sin(8x))^5 = 5(\cos(\sin(8x))^4 \cdot \frac{d}{dx}(\cos(\sin(8x)) = 5(\cos(\sin(8x))^4 \cdot -\sin(\sin(8x)) \cdot \frac{d}{dx}(\sin(8x)) = -5(\cos(\sin(8x))^4 \cdot \sin(\sin(8x)) \cdot 8 \cos(8x)$$
$$= -40(\cos(\sin(8x))^4 \cdot \sin(\sin(8x)) \cdot \cos(8x) = -40\cos^4(\sin(8x)) \cdot \sin(\sin(8x)) \cdot \cos(8x)$$

Or is your answer somehow equivalent to this?

 

[Mark44]

Wait, isn't it

$$f'(x) = -40\cos^4(\sin(8x)) \cdot \sin(\sin(8x)) \cdot \cos(8x)$$
?

No. You should not have a factor of sin(sin(8x)).

The original function is a composite of three functions: g(h(k(x))), where g(x) = x⁵, h(x) = sin(x), and k(x) = 8x.

f'(x) = g'(h(k(x)) * h'(k(x)) * k'(x)

Because

$$\frac{d}{dx} (\cos(\sin(8x))^5 = 5(\cos(\sin(8x))^4 \cdot \frac{d}{dx}(\cos(\sin(8x)) = 5(\cos(\sin(8x))^4 \cdot -\sin(\sin(8x)) \cdot \frac{d}{dx}(\sin(8x)) = -5(\cos(\sin(8x))^4 \cdot \sin(\sin(8x)) \cdot 8 \cos(8x)$$
$$= -40(\cos(\sin(8x))^4 \cdot \sin(\sin(8x)) \cdot \cos(8x) = -40\cos^4(\sin(8x)) \cdot \sin(\sin(8x)) \cdot \cos(8x)$$

Or is your answer somehow equivalent to this?

 

[spamiam]

Okay, either I'm being really thick or one of us is misreading the question.

Homework Statement

What is the derivative of f(x) = cos^5(sin(8x))

f(x) = cos^5(sin(8x)) = (cos(sin(8x)))^5.

Then f is the composition of 4 functions namely A(x) = x⁵, B(x) = cos(x), C(x) = sin(x), and D(x) = 8x.

Then f(x) = A(B(C(D(x)))).

And just to further persuade you, see attached.