Complete the table without using the Quotient Rule.
Function
y=2x^(7/2)/x
Rewrite y=
Differentiate y'=
Simplify y' =
since x^n/x = x^(n-1),
y = 2x^(7/2)/x = 2x^(5/2)
y' = 2(5/2)x^(3/2) = 5x√x
To complete the table without using the Quotient Rule, we can rewrite the function and then differentiate it using the Power Rule.
1. Rewriting the function:
The given function is y = 2x^(7/2)/x. To write it in a form suitable for differentiation using the Power Rule, we can simplify the expression by canceling out the common factor in the numerator and denominator.
y = 2x^(7/2)/x
= 2x^(7/2 - 1)
Simplifying further:
= 2x^(5/2)
Therefore, the rewritten function is y = 2x^(5/2).
2. Differentiating the rewritten function:
Now, we can differentiate the function y = 2x^(5/2) using the Power Rule, which states that d/dx (x^n) = n * x^(n-1).
Applying the Power Rule, we get:
y' = d/dx (2x^(5/2))
= 2 * d/dx (x^(5/2))
= 2 * (5/2) * x^(5/2 - 1)
Simplifying further:
= 2 * (5/2) * x^(3/2)
Therefore, the derivative of y with respect to x (y') is given by y' = 5x^(3/2).
3. Simplifying the derivative:
The derivative, y' = 5x^(3/2) is already in its simplified form, so no further simplification is required in this step.
Hence, the completed table is as follows:
Function y = 2x^(5/2)
Differentiate y' = 5x^(3/2)
Simplify y' = 5x^(3/2)