Problem 18 - Calculus Single Variable

Question

Differentiate the given expression with respect to \(x\). \(x^{-5} e^{x}\)

Step-by-Step Solution

Answer

The derivative of \(x^{-5} e^x\) with respect to \(x\) is \(e^x x^{-6}(-5 + x)\).

Step 1: Identify the Function Components

We start by identifying the components of the function we need to differentiate. The function is the product of two functions: the power function \(x^{-5}\) and the exponential function \(e^x\).

Step 2: Apply the Product Rule

The expression \(x^{-5} e^x\) requires the use of the product rule for differentiation, which states that if you have two functions \(u(x)\) and \(v(x)\), then \(\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\). Here, let \(u(x) = x^{-5}\) and \(v(x) = e^x\).

Step 3: Differentiate \(u(x) = x^{-5}\)

Differentiate \(u(x) = x^{-5}\) with respect to \(x\). Using the power rule, the derivative \(u'(x) = -5x^{-6}\).

Step 4: Differentiate \(v(x) = e^x\)

The derivative of \(v(x) = e^x\) with respect to \(x\) is simply \(v'(x) = e^x\), as the derivative of \(e^x\) is itself.

Step 5: Substitute into the Product Rule Formula

Using the product rule, substitute the derivatives obtained into the formula: \(u'(x)v(x) + u(x)v'(x) = (-5x^{-6})(e^x) + (x^{-5})(e^x)\).

Step 6: Simplify the Expression

Simplify the expression by factoring out the common factor of \(e^x x^{-6}\):\[-5x^{-6}e^x + x^{-5}e^x = e^x x^{-6}(-5 + x).\]

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which essentially tells us how the function changes at any given point. This is vital in understanding how quantities vary in relation to each other.
The derivative is symbolic of the slope of a tangent line at any given point on the function's graph. When you differentiate a function, you determine this rate of change.
In the context of the exercise, we needed to differentiate the product of two functions: the power function, which in this case is a polynomial, and the exponential function, which involves the natural constant, e.

Power Rule

The Power Rule is a key tool in differentiation, especially for polynomial functions. It provides a straightforward method for finding the derivative of any term that is a power of x.
Simply put, if you have a function in the form of \[f(x) = x^n\],then its derivative will be:\[f'(x) = nx^{n-1}\].
In the given exercise, we applied the Power Rule to the function\[u(x) = x^{-5}\].
This gave us the derivative\[u'(x) = -5x^{-6}\]. This rule makes it easy to handle various powers of x, simplifying the differentiation process significantly.

Exponential Function

Exponential functions are a remarkable category of functions where the variable appears in the exponent. The most common exponential function is based on Euler's number, e, and is expressed as \(f(x) = e^x\).
This function is unique because its derivative is the same as the original function: \[f'(x) = e^x\].
For our exercise, we had the function \[v(x) = e^x\], whose derivative \[v'(x) = e^x\]. The simplicity of its differentiation makes exponential functions extremely important in various scientific and engineering fields.

Calculus

Calculus is the branch of mathematics that studies continuous change, typically divided into two main areas: differentiation and integration.
Differentiation, the focus of our current exercise, deals with the concept of change and rates of change. It's the process of finding a derivative, as we've applied using the product and power rules.
Calculus offers powerful techniques for both theoretical developments and practical problem-solving. By understanding both differentiation and integration, one can analyze and predict changes in many fields, from physics to economics. This makes calculus a pivotal tool in both academia and industry.