Question

Find the derivative of the function. $$ g(x)=\sqrt{x}-3 e^{x} $$

Short answer

The derivative of the function \(g(x) = \sqrt{x} - 3e^{x}\) is \(g'(x)= \frac{1}{2\sqrt{x}} - 3e^{x}\)

Step-by-step solution

Apply Power Rule

For the first term \(\sqrt{x}\), rewrite it in exponential form as \(x^{1/2}\). Apply the Power Rule, which says that the derivative of \(x^n\) is \(nx^{n-1}\). This derives into \(\frac{1}{2}x^{-1/2}\) or \(\frac{1}{2\sqrt{x}}\).

Apply Exponential Rule

For the second term \(3e^{x}\), apply the exponential rule which states that the derivative of \(e^{ax}\) is \(ae^{ax}\). Differentiation leads to \(3e^{x}\). Note that the exponent 3 remains the same since the derivative of \(e^{x}\) is \(e^{x}\).

Combine the Derivatives

Combine the derivatives of each term to get the final derivative of the function: \(g'(x)= \frac{1}{2\sqrt{x}} - 3e^{x}\)

Key concepts

Power Rule

When starting off with calculus, one of the first tools you'll need is the differentiation's Power Rule. It's beautifully straightforward and is used to find the derivative of a power function where the variable, usually denoted as 'x', is raised to a constant exponent.

Imagine you're working with the function \( f(x) = x^n \), where \(n\) is any real number. The Power Rule states that to get the derivative \(f'(x)\), you simply multiply the exponent \(n\) with the base \(x\), and then subtract one from the exponent. This gives us a new function \(f'(x) = nx^{n-1}\).

Apply this rule to \(\sqrt{x}\), which can be rewritten as \(x^{1/2}\). Using the Power Rule yields \(\frac{1}{2}x^{-1/2}\) or equivalently, \(\frac{1}{2\sqrt{x}}\). This technique turns something that might seem daunting, like square roots, into a more manageable form for differentiation.

Exponential Rule

Next, let's demystify the Exponential Rule for derivatives. This rule comes into play when you're working with the natural exponential function, \(e^x\).

Under the Exponential Rule, the derivative of an exponential function like \(e^{ax}\), where \(a\) is a constant, is simply the original function multiplied by the constant. Thus, the derivative of \(e^{ax}\) is \(ae^{ax}\). This might seem magical at first, but it's a property unique to our friend \(e\).

In practice, if your original function is \(3e^x\), differentiation is a breeze. The derivative, according to the Exponential Rule, is the constant (3) times the derivative of \(e^x\), which is \(e^x\) itself. So, you end up with \(3e^x\) once again, no change in the exponent because of the special exponential characteristic.

Differentiation

At its core, differentiation is a fundamental process in calculus used to calculate the rate at which something changes. When you differentiate a function, you're finding its derivative, which provides the slope of the function at any point along its curve.

The derivative tells us how quickly or slowly the function's output (usually \(y\) or \(f(x)\)) changes as its input (usually \(x\)) changes. This concept is vital for understanding motion, growth, and many other dynamic processes.

In the context of the exercise, you differentiate the function \(g(x)=\sqrt{x}-3e^{x}\) by applying the Power Rule to the first term and the Exponential Rule to the second term. After differentiating both terms, you combine them to get the final derivative \(g'(x) = \frac{1}{2\sqrt{x}} - 3e^{x}\). This represents the instantaneous rate of change of the function \(g(x)\) for any value of \(x\).