Question

Finding a Pattern Consider the function \(f(x)=g(x) h(x)\) (a) Use the Product Rule to generate rules for finding \(f^{\prime \prime}(x)\) , \(f^{\prime \prime \prime}(x),\) and \(f^{(4)}(x)\) . (b) Use the results of part (a) to write a general rule for \(f^{(n)}(x)\)

Short answer

The general rule for the nth derivative, \(f^{(n)}(x)\), of the function \(f(x) = g(x)h(x)\) is given by the nth row of Pascal's Triangle, where each term consists of a coefficient from Pascal's triangle, g and h to complementary powers, and a derivative of the same order of the power.

Step-by-step solution

Applying the Product Rule for the First Derivative

First, find the derivative \(f'(x)\) of \(f(x) = g(x)h(x)\) using the Product Rule. The Product Rule states that the derivative of the product of two functions is the derivative of the first times the second function plus the first function times the derivative of the second. So, \(f'(x) = g'(x)h(x) + g(x)h'(x)\).

Applying the Product Rule for the Second Derivative

Next, find the second derivative \(f''(x)\). This is done by again applying the Product Rule, but this time to each term of \(f'(x)\). This gives: \(f''(x) = g''(x)h(x) +2g'(x)h'(x) + g(x)h''(x)\).

Applying the Product Rule for the Third Derivative

Now, find the third derivative \(f'''(x)\). Once more, apply the Product Rule to each term of \(f''(x)\), which gives: \(f'''(x) = g'''(x)h(x) + 3g''(x)h'(x) + 3g'(x)h''(x) + g(x)h'''(x)\).

Identifying a Pattern

The structure of the derivatives suggests there's a pattern emerging. Each derivative consists of terms that multiply derivatives of g and h, where the sum of the indices of the derivatives equals the order of the derivative. The first, second and third derivatives of \(f(x)\) each show coefficients that occur in the binomial expansion, which are derived from the Binomial theorem. So the fourth derivative is likely to follow this pattern.

Using the Pattern to Write a General Rule

The pattern observed suggests that each term in the nth derivative is a term in the binomial expansion of \((g+h)^n\). The general rule for the nth derivative, \(f^{(n)}(x)\), of the function \(f(x) = g(x)h(x)\) is given by the nth row of Pascal's Triangle. Each term consists of a coefficient from Pascal's triangle, g and h to complementary powers, and a derivative of the same order of the power.

Key concepts

Second Derivative

When you hear about the second derivative, think of it as the derivative of a derivative. For a function, the first derivative represents the rate of change. Now, go a step further. The second derivative tells you how that rate of change is changing. It's like checking how fast the speed of your car is increasing or decreasing, not just how fast you're driving.

Let's apply this concept using the Product Rule from calculus. Suppose you have a function composed of two multiplied functions, say \(f(x) = g(x) \times h(x)\). The first step is to find the first derivative, \(f'(x)\). Using the Product Rule, you get \(f'(x) = g'(x)h(x) + g(x)h'(x)\).

To get the second derivative, labelled \(f''(x)\), apply the Product Rule again to each term in \(f'(x)\). You'll find that \(f''(x) = g''(x)h(x) + 2g'(x)h'(x) + g(x)h''(x)\). Notice how the same rule is used, but now derivatives of \(g\) and \(h\) become even more interwoven.

Third Derivative

The third derivative is like an extension of the concept of the second derivative. After understanding the rate at which the speed changes (the acceleration), you want to understand how that acceleration is changing. This change of acceleration is what the third derivative reflects.

To calculate the third derivative of a product of functions, again apply the Product Rule. Start from the second derivative \(f''(x)\), then apply the Product Rule successively: \(f'''(x) = g'''(x)h(x) + 3g''(x)h'(x) + 3g'(x)h''(x) + g(x)h'''(x)\).

Look at how the coefficients 1, 3, 3, and 1 appear here. They resemble entries from Pascal's Triangle, hinting at a deeper pattern. This is essential for understanding higher order derivatives and general rules that may emerge as fun mathematical patterns appear.

Binomial Theorem

The Binomial Theorem offers an elegant way of expanding expressions of the form \( (a + b)^n \). If you take a closer look at how the derivatives were structured in prior sections, you may notice a connection to binomial coefficients that arise in the binomial expansion.

When calculating higher derivatives like \(f'''(x)\), the coefficients (1, 3, 3, 1) align with those in the expansion of \( (g+h)^3 \) using the Binomial Theorem:\

• The first term: \(1 \ imes g^3 \ imes h^0\)
• The second term: \(3 \ imes g^2 \ imes h^1)\)
• And so on...

These coefficients are the same as those found in rows of Pascal's Triangle, and they help in calculating nth order derivatives efficiently.

Pascal's Triangle

Pascal's Triangle is a simple triangular array of numbers that spelling a wealth of algebra's wonders. Each row contains the coefficients for the expansion of a binomial expression \( (a+b)^n \).

This triangle is built such that each number is the sum of the two directly above it. A critical observance is how these coefficients appear in the sequences of derivatives. For the nth derivative of a product function, the coefficients seen align with those in Pascal's Triangle. This is what the structure of derivatives like \(f'''(x)\) shows, with coefficients 1, 3, 3, 1 matching entries in the third row of this triangle.

Understanding this alignment assists you in visualizing derivative structures for any complex function built as a product, making it as easy and as intuitive as interpreting patterns in this famous triangle.