Differentiate sin x - x cos x

panderson

1 year ago

So Its Me Anonymous time for me to solve this:
We need to differentiate: 𝑦 = sin 𝑥 − 𝑥cos𝑥
Step 1: Differentiate Term by Term

Using the derivative rules:
Derivative of sin𝑥:
𝑑y/𝑑𝑥(sin⁡𝑥)=cos⁡𝑥
Derivative of 𝑥cos⁡𝑥 (Product Rule): The product rule states:
𝑑y/𝑑𝑥[𝑢𝑣]=𝑢′𝑣+𝑢𝑣′
Let 𝑢 = 𝑥 and 𝑣 = cos𝑥,
then: 𝑢′ = 𝑑y/𝑑𝑥 𝑥=1
𝑣′ = 𝑑y/𝑑𝑥 cos𝑥 = −sin𝑥
Applying the product rule:
𝑑y/𝑑𝑥(𝑥cos⁡𝑥)=(1)(cos⁡𝑥)+(𝑥)(−sin𝑥) = cos𝑥 − 𝑥sin𝑥
Step 2: Put Everything Together
𝑑𝑦𝑑𝑥=cos⁡𝑥−(cos⁡𝑥−𝑥sin⁡𝑥)
Distribute the negative sign:
𝑑𝑦𝑑𝑥 = cos⁡𝑥 − cos𝑥 + 𝑥sin𝑥
​
Final Answer:
𝑑𝑦𝑑𝑥=𝑥sin𝑥

Leinad321

2 years ago

sinx when differentiated gives cosx
gor xcos x
let u=x and v=cos x
du/dx=1 dv/dx=-sinx (differentiating individually first)

using the product rule i.e
dy/dx=du/dx ×V +dv/dx ×U
dy/dx=1×cos x+(-sinx × x)
dy/dx=cosx-xsinx
put back in equation
cosx-(cosx-xsinx)
cosx-cosx+xsinx
=xsinx

runaboy

1 year ago

i don't understand