Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of \(\sqrt{x}\). $$f(x)=3 e^{x}+5 x+5$$

Short answer

Question: Find the derivative of the function $$f(x) = 3e^x + 5x + 5$$. Answer: The derivative of the function is $$f'(x) = 3e^x + 5$$.

Step-by-step solution

Understand the function

We have the function $$f(x)=3 e^{x}+5 x+5$$. We will differentiate this function with respect to x, which means finding its derivative. The derivative represents the rate of change of the function at any given point.

Apply the constant rule

The constant rule states that the derivative of a constant is zero. In our function, the constant is 5, so we have: $$\frac{d(5)}{dx} = 0$$

Apply the power rule

The power rule states that the derivative of a function $$x^n$$ is given by $$nx^{n-1}$$. In our function, we have $$5x$$, so the derivative will be: $$\frac{d(5x)}{dx} = 5\frac{d(x)}{dx} = 5(1) = 5$$

Apply the derivative of the exponential function

The derivative of the exponential function $$e^x$$ is itself, which means: $$\frac{d(e^x)}{dx} = e^x$$ In our function, we have $$3e^x$$, so the derivative is: $$\frac{d(3e^x)}{dx} = 3\frac{d(e^x)}{dx} = 3e^x$$

Combine the derivatives

Now we have found the derivatives of all the individual terms in the function. We will combine them to find the derivative of the entire function: $$\frac{d}{dx}(3e^x + 5x + 5) = \frac{d(3e^x)}{dx} + \frac{d(5x)}{dx} + \frac{d(5)}{dx}$$ $$f'(x) = 3e^x + 5 + 0$$ $$f'(x) = 3e^x + 5$$ The derivative of the given function $$f(x)=3 e^{x}+5 x+5$$ is $$f'(x) = 3e^x + 5$$.

Key concepts

Power Rule

The Power Rule is a fundamental concept in calculus used to find the derivative of polynomial functions. It's a handy rule where for any function of the form \(x^n\), the derivative is given by \(nx^{n-1}\). Essentially, you multiply by the power and reduce the power by one. This rule simplifies the process of differentiation, allowing you to solve for derivatives quickly.
For instance, consider the term \(5x\), as seen in our exercise. Here, the power \(n\) is 1, since \(x\) is the same as \(x^1\). Applying the Power Rule, you get:

• Multiply the coefficient by the power: \(5 \times 1 = 5\)
• Decrease the power of \(x\) by one: \(x^{1-1} = x^0 = 1\)

So, the derivative of \(5x\) is \(5\). This makes differentiation straightforward when dealing with polynomial expressions, helping to break down more complex functions into manageable pieces.

Exponential Function

Exponential functions, particularly those involving \(e\), the natural exponential base, are crucial in calculus. The derivative of \(e^x\) is uniquely interesting because it is equal to itself: \(\frac{d(e^x)}{dx} = e^x\). This property makes them very predictable in terms of derivatives.

Consider the term \(3e^x\) from our exercise problem. Here, it's important to recognize that taking the derivative involves using the constant multiplied by the derivative of the exponential function itself. When differentiating \(3e^x\), you carry the constant 3, giving:

• Differentiate \(e^x\) to get \(e^x\)
• Multiply by the constant: \(3 \times e^x = 3e^x\)

This consistency and simplicity of exponential function derivatives make them particularly straightforward to work with in calculus, allowing you to predict the rate of change of exponential growth or decay scenarios easily.

Constant Rule

The Constant Rule is one of the simplest rules in calculus. It states that the derivative of a constant is zero. This stems from the idea that constants have no rate of change; they are constant in value, so their contribution to a derivative is nullified.

In the context of our problem, look at the constant term \(5\). According to the Constant Rule, its derivative is:

• Derivative of the constant \(5\) is \(0\)

By applying this rule, we can ignore constant terms when calculating derivatives as their contribution is zero. This helps simplify expressions and focus on terms that actually affect the function's rate of change.