91Ó°ÊÓ

In calculus, finding the derivative of a function is an essential process. The derivative of a function measures how the function's output changes as its input changes. This is often denoted as \(\frac{dy}{dx}\), which indicates the rate of change of \(y\) with respect to \(x\).

To calculate the derivative, you apply differentiation rules to each part of the function separately. In our function \(y = 3x^2 + e^x\), we need to differentiate both \(3x^2\) and \(e^x\).

• The derivative of a power function like \(x^2\) is \(2x\), and when it is multiplied by 3, it becomes \(3 \cdot 2x = 6x\).
• The derivative of the exponential function \(e^x\) is simply \(e^x\).

Together, we combine these derivatives to find the overall derivative of the function, which is \(\frac{dy}{dx} = 6x + e^x\). After finding this expression, you can use it to evaluate the function at particular points.

One of the powerful tools in differentiation is the sum rule. The sum rule allows you to calculate the derivative of a sum of functions by differentiating each function separately and then adding them together.

Within our given function \(y = 3x^2 + e^x\), we see it is a sum of \(3x^2\) and \(e^x\). The sum rule of differentiation tells us that we can find the derivative of this by calculating each component individually with respect to the variable \(x\) and then summing them.

• First, differentiate \(3x^2\), yielding \(6x\).
• Then, differentiate \(e^x\), yielding \(e^x\).

Adding these together gives us the total derivative: \(\frac{dy}{dx} = 6x + e^x\). This approach simplifies the process of breaking down complex functions into manageable pieces, making it easier to handle more complicated problems.

Once you have found the expression for a derivative, evaluating it at a specific point allows you to determine the rate of change at that exact moment. This gives you insight into the behavior of the function at a particular value of \(x\).

In our example, we need to evaluate the derivative \(\frac{dy}{dx} = 6x + e^x\) when \(x = 0.5\). To do this, simply substitute \(0.5\) into the derivative.

• Calculate \(6 \times 0.5 = 3\).
• Compute \(e^{0.5}\), which is a numerical calculation for the exponential function.
• Add them together: \(3 + e^{0.5}\).

Thus, the rate of change of the function \(y = 3x^2 + e^x\) at \(x = 0.5\) is \(3 + e^{0.5}\). This tells us how steeply the function is increasing at that specific point on its curve.