Verify the Sum and Product Rules for derivatives of vector-valued functions.

Question

SaurbHurbkj · Accepted Answer

Step 1To Determine: Verify the Sum and Product Rules for derivatives of vector-valued functions.Given: we have a vector valued function.Explanation: we know that the property of sum and for production of the vector valued function:property: the derivative of the sum of two vector valued function is equal to the sum of each functions derivative.(F→(t)+G→(t))′=F′→(t)+G′→(t)now let us consider an exampleF→(t)=(t2,5t) and G→(t)=(sin⁡t,cos⁡t)now finding L.H.S we have the sum equals to(F→(t)+G→(t))=(t2+sin⁡t,5t+cos⁡t)now the derivative is(F→(t)+G→(t))′=(2t+cos⁡t,5−sin⁡t)now finding R.H.S we have derivativeF′→(t)=(2t,5)G′→(t)=(cos⁡t−sin⁡t)F′→(t)+G′→(t)=(2t+cos⁡t,5−sin⁡t)hence LHS=RHSStep 2property: if you multiply a scalar function by a vector values function the derivative of their product will follows the product rule for derivative(f(t)G→(t))′=f(t)G→(t)+G′→(t)f(t)let us considerf(t)=t2 and G→(t)=(t4,ln⁡t)f(t)G→(t)=(t6,t2ln⁡t)[f(t)G→(t)]′=(6t5,2tln⁡t+t)now the RHS will bef(t)=t2 and G→(t)

Verify the Sum and Product Rules for derivatives of vector-valued

Answered question

Answer & Explanation

New Questions in Calculus 2