Find the differential of the function.
[tex]$d y=\frac{1}{x^7-8 x}$[/tex]
Answer (1)
Rewrite the function as y = ( x 7 − 8 x ) − 1 .
Apply the chain rule to find the derivative d x d y = ( x 7 − 8 x ) 2 8 − 7 x 6 .
Multiply the derivative by d x to find the differential d y .
The differential of the function is d y = ( x 7 − 8 x ) 2 8 − 7 x 6 d x .
Explanation
Problem Analysis We are given the function y = x 7 − 8 x 1 and we want to find its differential d y . The differential is related to the derivative by the formula d y = d x d y d x . So, we need to find the derivative of y with respect to x .
Applying the Chain Rule To find the derivative of y = x 7 − 8 x 1 , we can rewrite it as y = ( x 7 − 8 x ) − 1 . Now we can apply the chain rule. The chain rule states that if we have a function y = f ( g ( x )) , then d x d y = f ′ ( g ( x )) g ′ ( x ) . In our case, f ( u ) = u − 1 and g ( x ) = x 7 − 8 x .
Finding the Derivatives First, let's find the derivative of f ( u ) = u − 1 with respect to u : f ′ ( u ) = − 1 ⋅ u − 2 = − u − 2 . Next, let's find the derivative of g ( x ) = x 7 − 8 x with respect to x : g ′ ( x ) = 7 x 6 − 8 .
Calculating the Derivative Now, we can apply the chain rule: d x d y = f ′ ( g ( x )) ⋅ g ′ ( x ) = − ( x 7 − 8 x ) − 2 ⋅ ( 7 x 6 − 8 ) = ( x 7 − 8 x ) 2 − ( 7 x 6 − 8 ) = ( x 7 − 8 x ) 2 8 − 7 x 6 .
Finding the Differential Finally, we can find the differential d y by multiplying the derivative by d x : d y = d x d y d x = ( x 7 − 8 x ) 2 8 − 7 x 6 d x .
Final Answer Thus, the differential of the function y = x 7 − 8 x 1 is d y = ( x 7 − 8 x ) 2 8 − 7 x 6 d x .
Examples
In physics, if you have a quantity (like the electric potential) that depends on position according to the function y = x 7 − 8 x 1 , the differential d y tells you how much the electric potential changes when you move a tiny distance d x . This is useful for calculating electric fields or forces.
