Question

In Exercises \(1-28\) , find \(d y / d x\) . Remember that you can use NDER to support your computations. $$y=e^{2 x / 3}$$

Short answer

So, the derivative \(d y / d x\) of the function \(y=e^{2 x / 3}\) is \(\frac{2}{3} e^{\frac{2 x}{3}}\).

Step-by-step solution

Identify the Outer and Inner Functions

Identify the outer and inner functions. Here, the outer function is \(e^x\) and the inner function is \(\frac{2x}{3}\). Now, it is necessary to apply the chain rule.

Application of Chain Rule

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. The chain rule formula is: \((f(g(x)))' = f'(g(x)) \cdot g'(x)\). Applying this: Take the derivative of the outer function: \(e^x\) which is still \(e^x\). But instead of just 'x', replace 'x' with the inner function \(\frac{2x}{3}\). So, we get \(e^{\frac{2x}{3}}\).

Derivative of Inner Function

Next, find the derivative of the inner function \(\frac{2x}{3}\). Using the power rule, the derivative of \(\frac{2x}{3}\) is \(\frac{2}{3}\).

Multiply Derivatives of Inner and Outer Functions

Finally, multiply the derivative of the outer function by the derivative of the inner function as required by the chain rule: \(e^{\frac{2x}{3}} \cdot \frac{2}{3} = \frac{2}{3} e^{\frac{2x}{3}}\).