Find [tex]\frac{d y}{d x}[/tex] of the following [tex]y=\frac{\sin 5 x}{5 x^4}[/tex]
Answer (1)
Apply the quotient rule: d x d ( v u ) = v 2 v u ′ − u v ′ , where u = sin 5 x and v = 5 x 4 .
Find the derivatives: u ′ = 5 cos 5 x and v ′ = 20 x 3 .
Substitute into the quotient rule: d x d y = ( 5 x 4 ) 2 ( 5 x 4 ) ( 5 c o s 5 x ) − ( s i n 5 x ) ( 20 x 3 ) .
Simplify the expression: d x d y = 5 x 5 5 x cos 5 x − 4 sin 5 x .
Explanation
Problem Analysis We are given the function y = 5 x 4 s i n 5 x and we want to find its derivative with respect to x , which is denoted as d x d y . This requires us to use the quotient rule.
Quotient Rule The quotient rule states that if y = v ( x ) u ( x ) , then d x d y = [ v ( x ) ] 2 v ( x ) ⋅ u ′ ( x ) − u ( x ) ⋅ v ′ ( x ) . In our case, u ( x ) = sin 5 x and v ( x ) = 5 x 4 .
Derivative of u(x) First, we find the derivative of u ( x ) = sin 5 x with respect to x . Using the chain rule, we have d x d u = 5 cos 5 x .
Derivative of v(x) Next, we find the derivative of v ( x ) = 5 x 4 with respect to x . Using the power rule, we have d x d v = 20 x 3 .
Applying Quotient Rule Now, we substitute u ( x ) , v ( x ) , d x d u , and d x d v into the quotient rule formula: d x d y = ( 5 x 4 ) 2 ( 5 x 4 ) ( 5 cos 5 x ) − ( sin 5 x ) ( 20 x 3 )
Simplifying Simplify the expression: d x d y = 25 x 8 25 x 4 cos 5 x − 20 x 3 sin 5 x
Further Simplification Further simplification by factoring out 5 x 3 from the numerator: d x d y = 25 x 8 5 x 3 ( 5 x cos 5 x − 4 sin 5 x ) d x d y = 5 x 5 5 x cos 5 x − 4 sin 5 x
Final Answer Thus, the derivative of y with respect to x is: d x d y = 5 x 5 5 x cos 5 x − 4 sin 5 x
Examples
In physics, if y represents the displacement of a particle as a function of time x , then d x d y gives the velocity of the particle. For example, if the displacement is given by y = 5 x 4 s i n 5 x , finding the derivative allows us to determine the velocity of the particle at any given time x . This is crucial in analyzing the motion of objects in various physical systems.
