Directions: For each category see if you can devise the rule(s) that was used to

Question

Directions: For each category see if you can devise the rule(s) that was used to find the derivative(s) and then use it to find the derivatives of the functions that follow.

Category 1: Sine and Cosine

Function: f(x)=sin(2x) 3. Function: f(x)=3sin(x^2)

Derivative: f^' (x)=2cos(2x) Derivative: f^' (x)=6xcos(x^2)

Function: f(x)=cos(5x) 4. Function: f(x)=cos(x^3)

Derivative: f^' (x)=-5sin(5x) Derivative: f^' (x)=-3x^2 sin(x^3)

What is the formula for the derivative of Sine? What about for Cosine? What role does the chain rule play in these formulas?

Find the derivatives of the following:

f(x)=2sin?(3x-2) 2. f(x)=-cos(x^2-5x)

Explanation / Answer

Derivative of sin x = cos x Derivative of cosine x= - sin x And here chain rule is using as in 1st problem, we did derivativr of sin 2x = cos 2x and then derivative of 2x is 2. So, final answer is 2 cos 2x and similarly applied for other problems. In finding derivatives, f(x) = 2sin^n(3x-2) Assuming? = n we get f'(x) = 2n sin^(n-1)(3x-2) * cos(3x-2)*3 2.) f(x) = -cos(x^2 - 5x) f'(x) = sin(x^2 - 5x)*(2x-5)

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