Find the derivative of y = 2x cos x – 2 sin x Please show all work.
The Correct Answer and Explanation is:
To find the derivative of the function y=2xcosx−2sinxy = 2x \cos x – 2 \sin xy=2xcosx−2sinx, we will apply the rules of differentiation, specifically the product rule and the standard derivatives of trigonometric functions.
Step 1: Identify the components
The function is y=2xcosx−2sinxy = 2x \cos x – 2 \sin xy=2xcosx−2sinx. This is the sum of two terms, so we will differentiate each term separately.
Step 2: Differentiate the first term 2xcosx2x \cos x2xcosx
This term is a product of two functions, 2x2x2x and cosx\cos xcosx. To differentiate a product, we use the product rule: ddx(u⋅v)=u′v+uv′\frac{d}{dx}(u \cdot v) = u’v + uv’dxd(u⋅v)=u′v+uv′
Here, let u=2xu = 2xu=2x and v=cosxv = \cos xv=cosx. Now, differentiate each part:
- The derivative of u=2xu = 2xu=2x is u′=2u’ = 2u′=2.
- The derivative of v=cosxv = \cos xv=cosx is v′=−sinxv’ = -\sin xv′=−sinx (using the standard derivative of cosx\cos xcosx).
Applying the product rule: ddx(2xcosx)=(2)⋅(cosx)+(2x)⋅(−sinx)=2cosx−2xsinx\frac{d}{dx}(2x \cos x) = (2) \cdot (\cos x) + (2x) \cdot (-\sin x) = 2 \cos x – 2x \sin xdxd(2xcosx)=(2)⋅(cosx)+(2x)⋅(−sinx)=2cosx−2xsinx
Step 3: Differentiate the second term −2sinx-2 \sin x−2sinx
This is a simple trigonometric function. The derivative of sinx\sin xsinx is cosx\cos xcosx, so: ddx(−2sinx)=−2cosx\frac{d}{dx}(-2 \sin x) = -2 \cos xdxd(−2sinx)=−2cosx
Step 4: Combine the derivatives
Now, we combine the derivatives of the two terms: dydx=(2cosx−2xsinx)−2cosx\frac{dy}{dx} = (2 \cos x – 2x \sin x) – 2 \cos xdxdy=(2cosx−2xsinx)−2cosx
Simplifying the expression: dydx=2cosx−2xsinx−2cosx\frac{dy}{dx} = 2 \cos x – 2x \sin x – 2 \cos xdxdy=2cosx−2xsinx−2cosx
The 2cosx2 \cos x2cosx terms cancel out: dydx=−2xsinx\frac{dy}{dx} = -2x \sin xdxdy=−2xsinx
Final Answer:
dydx=−2xsinx\frac{dy}{dx} = -2x \sin xdxdy=−2xsinx
Explanation:
The derivative of y=2xcosx−2sinxy = 2x \cos x – 2 \sin xy=2xcosx−2sinx is found by using the product rule for the first term and the standard derivative for the second term. After simplifying, we get the result dydx=−2xsinx\frac{dy}{dx} = -2x \sin xdxdy=−2xsinx. This shows how we break down a more complex function into manageable pieces and apply differentiation rules step by step.
