Problem 34 - Calculus and its applications

Question

Differentiate each function. \(g(x)=5 x^{-3}\left(x^{4}-5 x^{3}+10 x-2\right)\)

Step-by-Step Solution

Answer

The derivative of the function is \(g'(x) = 5 - 100x^{-3} + 30x^{-4}\).

Step 1: Identify the Product Rule

The function given is the product of two functions: the first is a power function, \(5x^{-3}\), and the second is a polynomial, \(x^4 - 5x^3 + 10x - 2\). To differentiate products of functions, we will apply the product rule, which states: if \(h(x) = u(x)v(x)\), then \(h'(x) = u'(x)v(x) + u(x)v'(x)\).

Step 2: Differentiate \(u(x) = 5x^{-3}\)

To find \(u'(x)\), differentiate \(u(x) = 5x^{-3}\) using the power rule \(d/dx \, [x^n] = nx^{n-1}\). Thus, \(u'(x) = 5(-3)x^{-4} = -15x^{-4}\).

Step 3: Differentiate \(v(x) = x^4 - 5x^3 + 10x - 2\)

To find \(v'(x)\), differentiate each term using the power rule. For \(x^4\), the derivative is \(4x^3\); for \(-5x^3\), it is \(-15x^2\); for \(10x\), it is \(10\); and for the constant \(-2\), it is \(0\). Thus, \(v'(x) = 4x^3 - 15x^2 + 10\).

Step 4: Apply the Product Rule

Substitute the derivatives and the original functions into the product rule formula: \(h'(x) = u'(x)v(x) + u(x)v'(x)\). This gives: \(-15x^{-4}(x^4 - 5x^3 + 10x - 2) + 5x^{-3}(4x^3 - 15x^2 + 10)\).

Step 5: Simplify the Expression

Distribute \(-15x^{-4}\) through \(v(x)\) and \(5x^{-3}\) through \(v'(x)\), then combine like terms. \(-15x^{-4}(x^4 - 5x^3 + 10x - 2) = -15 + 75x^{-1} - 150x^{-3} + 30x^{-4}\), \(5x^{-3}(4x^3 - 15x^2 + 10) = 20 - 75x^{-1} + 50x^{-3}\). Combine:\(-15 + 75x^{-1} - 150x^{-3} + 30x^{-4} + 20 - 75x^{-1} + 50x^{-3} = 5 - 100x^{-3} + 30x^{-4}\).

Key Concepts

Understanding the Product Rule

When we are faced with a function that is a product of two simpler functions, we use the product rule to differentiate it. The product rule formula is crucial here. Given two functions, say \(u(x)\) and \(v(x)\), the derivative of their product \(h(x) = u(x)v(x)\) is expressed as:

• \( h'(x) = u'(x)v(x) + u(x)v'(x) \)

This rule tells us to find the derivative of the first function \(u(x)\), multiply it by the second function \(v(x)\), and then add to it the first function \(u(x)\) multiplied by the derivative of the second function \(v'(x)\). This approach ensures none of the parts are missed. In our exercise, the functions are \(5x^{-3}\) (our \(u(x)\)) and \(x^4 - 5x^3 + 10x - 2\) (our \(v(x)\)). Understanding this part is crucial because it sets the stage for how we'll proceed to take each derivative.

Exploring the Power Rule

The power rule is one of the simplest ways to differentiate a function. It's particularly useful for functions like \(x^n\), where \(n\) is any real number. The rule states:

• \( \frac{d}{dx} [x^n] = nx^{n-1} \)

Simply put, the exponent is brought down in front as a coefficient, and then the new exponent is decremented by one. For example, to differentiate \(5x^{-3}\), you would proceed as follows:

• Step 1: Bring down the exponent: \(-3\)
• Step 2: Multiply it by the coefficient: \(5\times (-3)\)
• Step 3: Decrease the exponent by one: \(-3-1 = -4\)

Thus, the derivative becomes \(-15x^{-4}\). This rule simplifies the differentiation process, allowing you to tackle each term of a polynomial one at a time, making it very manageable.

Mastering Polynomial Differentiation

Differentiating polynomials involves applying the power rule to each term separately. Here’s a breakdown for the polynomial \(x^4 - 5x^3 + 10x - 2\):

• For \(x^4\), the derivative is \(4x^3\).
• For \(-5x^3\), the derivative is \(-15x^2\).
• For \(10x\), the derivative is \(10\).
• The derivative of a constant \(-2\) is \(0\).

Each term is tackled one by one using the power rule, and the simplicity lies in repeating a process you are already familiar with. You then collect all these results: the derivative \(v'(x)\) becomes \(4x^3 - 15x^2 + 10\). The method reduces complexity by systematically addressing each element, allowing you to see how each contributes to the derivative of the whole function.