Question

Without doing any calculating, find each derivative. (a) \(D_{x}^{4}\left(3 x^{3}+2 x-19\right)\) (b) \(\quad D_{x}^{12}\left(100 x^{11}-79 x^{10}\right)\) (c) \(D_{x}^{11}\left(x^{2}-3\right)^{5}\)

Short answer

Derivatives in (a), (b), and (c) are all zero.

Step-by-step solution

Understanding Higher Derivatives (a)

In part (a), we're finding the 4th derivative of the polynomial \(3x^3 + 2x - 19\). Recall that taking derivatives repeatedly of terms like \(x^n\) will eventually lead the term to become zero if \(n\) is less than the number of derivatives taken. Since the highest degree here is a cubic term \(3x^3\), taking four derivatives will make all terms equal zero.

Analyzing Higher Derivatives (b)

In part (b), we have to find the 12th derivative of the expression \(100x^{11}-79x^{10}\). Applying similar logic, we know that repeatedly differentiating each term will eventually reduce them to zero. Specifically, a term like \(100x^{11}\) will become zero after the 11th derivative, and \(-79x^{10}\) will become zero after the 10th derivative. Therefore, taking the 12th derivative of any term in this polynomial will result in zero.

Applying the Chain Rule for Repeated Derivatives (c)

In part (c), we are asked for the 11th derivative of \((x^2 - 3)^5\). Utilizing the chain rule and the binomial expansion for repeated derivatives of a power of a polynomial, it becomes clear that after differentiating more times than the power itself, the result will be zero. Therefore, the 11th derivative of a polynomial to the 5th power is zero.

Key concepts

Polynomial Derivatives

Polynomial derivatives involve differentiating expressions that consist of terms like \(x^n\) where \(n\) is a non-negative integer. Taking derivatives of polynomials requires understanding how each term changes when differentiated.

The power rule is key here. It states that the derivative of \(x^n\) is \(nx^{n-1}\). This is because when you differentiate, the exponent \(n\) comes down as a factor, and you reduce the exponent by one. Thus, differentiating polynomial expressions is a matter of applying this rule to each term.

Repeated differentiation, or higher derivatives, focus on differentiating a polynomial multiple times. In exercises a and b, we see that after a certain number of derivatives, each term in the polynomial becomes zero.

For example, after differentiating a cubic term \(x^3\) four times, you would reduce through cubic, quadratic, linear, and eventually to zero. Similarly, for \(x^{11}\), after 12 differentiations, the result will be zero. Understanding this progression allows efficient solutions without lengthy calculations.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. A composite function is one where a function is inside another, such as \(f(g(x))\). The chain rule helps differentiate these complex situations by focusing on the derivative of the outer function and multiplying it by the derivative of the inner function.

In part (c) of our exercise, we are dealing with a function raised to a power, \((x^2 - 3)^5\). The chain rule states that if \(u(x) = (x^2 - 3)\), then the derivative of \(u^5\) requires differentiating \(u^5\) concerning \(u\), and then multiplying this by the derivative of \(u\) concerning \(x\).

This is why the formula \(\frac{d}{dx} (u^n) = n u^{n-1} \cdot \frac{du}{dx}\) is so useful. However, when asked for a higher derivative like the 11th, if the original power is only 5, the result will still be zero. The higher you differentiate, the sooner the result approaches zero, particularly if differentiating more than the initial power.

Derivative Techniques

Several derivative techniques are critical for efficiently solving calculus problems, particularly when working with polynomials and composite functions.

• Power Rule: Differentiating \(x^n\) results in \(nx^{n-1}\). Simplifying processes through this technique makes finding derivatives straightforward.
• Polynomial Rule: Each term is differentiated independently, and constants are carried along through the differentiation process.
• Chain Rule: Focuses on finding derivatives of nested functions, where you must break down the problem into outer and inner derivatives.

For problems involving higher derivatives, understanding repeated application of these techniques is crucial. Recognizing when entire terms drop out (become zero) after several derivatives simplifies efforts dramatically.

In tricky problems, these strategies promote efficiency and understanding, reducing computation while enriching your conceptual grasp. They equip you to handle even the most tedious derivatives.