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Differentiate
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Q. Differentiate
- Apply Chain Rule: Apply the chain rule to differentiate the function . The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
- Find : Let . Then . We will first find the derivative of with respect to , which is .
- Find : Now we need to find the derivative of with respect to , . To do this, we will use the quotient rule, which states that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the square of the denominator.
- Apply Quotient Rule: Differentiate the numerator and the denominator separately. The derivative of the numerator with respect to is , and the derivative of the denominator with respect to is .
- Simplify : Apply the quotient rule: .
- Find : Simplify the expression in the numerator: .
- Substitute : Now we have and , so we can find by multiplying these two derivatives: .
- Simplify : Substitute back into the equation to get in terms of : \frac{dy}{dx} = \(4\left(\frac{x+}{x}\right)^ \cdot \left(-\frac{}{(x)^}\right).
- Simplify : Substitute back into the equation to get in terms of : .Simplify the expression by combining the powers of : .
