Question

How do you find the derivative of `sqrtθ sin θ`?

Answer 1

Let `f(x)=sqrt(theta)sintheta` `=(theta)^(1/2)sintheta`
differentiate with respect to `theta` using product rule
`(df(x))/(d theta)=(d(theta)^(1/2))/(d theta)sintheta+theta^(1/2)(d(sintheta))/(d theta)`
`implies (df(x))/(d theta)=1/2theta^(1/2-1)sintheta+theta^(1/2)costheta`
`implies (df(x))/(d theta)=1/2theta^(-1/2)sintheta+sqrt(theta)costheta`
`implies (df(x))/(d theta)=1/2(1/theta^(1/2))sintheta+sqrt(theta)costheta`
`implies (df(x))/(d theta)=1/2(1/sqrt(theta))sintheta+sqrtthetacostheta`
`implies (df(x))/(d theta)=1/(2sqrttheta)sintheta+sqrtthetacostheta`