i don't know how to start this problem. i don't like cos and sin. im so confuse.
1.f(x)=sin(sin(x)), find f'(x)
2.f(x)= 4sin^4x, find f'(x). this problem i couldn't find it in my math book. i never see it before. because of sin exponent 4 -->sin^4
i got the result number 1 is cos(x)cos(sin(x)).
im still stuck with number 2.
cos(sin(x))*cos(x)
16sin^3(x)*cos(x)
d/dx(f(x))= f'(x)*x'
thanks byant
I understand that trigonometric functions like cosine and sine can be a bit confusing, but I'll try my best to explain how to approach these problems step by step.
1. To find the derivative of f(x) = sin(sin(x)), you can use the chain rule. The chain rule states that if you have a composition of two functions, such as f(g(x)), the derivative of this composition is the derivative of the outer function times the derivative of the inner function.
In this case, the outer function is sin(x), and the inner function is sin(x) as well. Let's call the inner function g(x).
First, find the derivative of the inner function g(x):
g'(x) = cos(x)
Next, find the derivative of the outer function f'(x) using the chain rule:
f'(x) = cos(g(x)) * g'(x)
Finally, substitute g(x) back in to get the final answer:
f'(x) = cos(sin(x)) * cos(x)
2. To find the derivative of f(x) = 4sin^4x, you are dealing with a power of sin(x). Here's how you can approach it:
Start by using a trigonometric identity: sin^2(x) = (1/2)(1 - cos(2x)). This identity can help you simplify the expression.
Rewrite f(x) as: f(x) = 4(sin^2(x))^2
Using the identity, substitute sin^2(x) with (1/2)(1 - cos(2x)):
f(x) = 4((1/2)(1 - cos(2x)))^2
Simplify the expression:
f(x) = 2(1 - cos(2x))^2
Now, you can differentiate f(x) using the power rule and the chain rule. Start by differentiating the outer function:
f'(x) = 2 * 2(1 - cos(2x))^1 * (-sin(2x)) * 2
Simplify the expression:
f'(x) = -8sin(2x)(1 - cos(2x))
So, the derivative of f(x) = 4sin^4x is f'(x) = -8sin(2x)(1 - cos(2x)).
Remember, practice makes perfect! Keep working on problems involving trigonometric functions, and you'll become more comfortable with them over time.