Question

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. $$g(r)=\left(5 r^{3}+3 r+1\right)\left(r^{2}+3\right)$$

Short answer

Answer: The derivative of the function g(r) is $$g'(r)=25r^4+54r^2+2r+9$$.

Step-by-step solution

Expand the product of polynomials

First, expand the expression: $$g(r)=\left(5 r^{3}+3 r+1\right)\left(r^{2}+3\right)$$ To expand, multiply each term in the first polynomial by each term in the second polynomial, and then combine like terms.

Combine like terms

After expanding the product, we get: $$g(r)=5r^5+3r^3+r^2+15r^3+9r+3$$ Combine the like terms: $$g(r)=5r^5+18r^3+r^2+9r+3$$

Find the derivative

Now we will use the power rule to find the derivative of the simplified expression. The power rule states that the derivative of a term with respect to a variable is obtained by multiplying the coefficient by the exponent, and then subtracting 1 from the exponent. Apply the power rule to each term in g(r): $$\frac{d}{dr}g(r)=\frac{d}{dr}(5r^5)+\frac{d}{dr}(18r^3)+\frac{d}{dr}(r^2)+\frac{d}{dr}(9r)+\frac{d}{dr}(3)$$

Apply power rule

Apply the power rule to each term: $$\frac{d}{dr}g(r)=25r^4+54r^2+2r+9$$ So, $$g'(r)=25r^4+54r^2+2r+9$$

Key concepts

Power Rule

When working with derivatives in calculus, the power rule is one of the most basic and useful rules you'll encounter. It applies to functions of the form f(x) = ax^n where a is a constant, n is a real number, and x is the variable. The derivative of this function with respect to x is f'(x) = nax^{n-1}. Essentially, you bring down the exponent as a coefficient and then subtract one from the exponent. This process simplifies the calculation of derivatives enormously, especially for polynomial functions.

Here's a real-world analogy to help understand it better: If you think of the exponent as a level of 'power' something has, the power rule tells us that to find the rate at which this power changes, you multiply the current level of power by the number of levels you have, then move down one level. It's like climbing a staircase – you start on step n and take a step down to n-1, but you also carry with you the power of the step you were just on.

In our exercise, applying the power rule to 5r^5 gives us 25r^4, because the exponent 5 is brought in front, and we then subtract one from the exponent to get 4. The constant multiplier, 5 in this case, stays the same.

Derivative of a Polynomial

A polynomial is a mathematical expression that consists of variables and constants combined using addition, subtraction, multiplication, and non-negative integer exponents. To find the derivative of a polynomial, which gives you the slope of the tangent line to the polynomial's curve at any point, you apply the power rule to each individual term.

In the context of our exercise, g(r) is a polynomial that first needed to be expanded. Once we expanded and combined like terms, we got a simpler polynomial, g(r)=5r^5+18r^3+r^2+9r+3. To find the derivative of this entire expression, we find the derivatives of each term separately using the power rule and then add them together.

For example, for the term 18r^3, applying the power rule means you multiply the exponent 3 by the coefficient 18 to get 54, and then decrease the exponent by one, resulting in a derivative of 54r^2. Repeat this process for all terms, and you get the final derivative, adding up all the individual derivatives from each term.

Combining Like Terms

Combining like terms is a critical step in simplifying mathematical expressions, especially when dealing with polynomials. Like terms are terms that have the same variables raised to the same power, even though they might have different coefficients. To combine them, simply add or subtract the coefficients while keeping the variable part unchanged.

In our exercise, after expanding the product of polynomials, we had terms like 3r^3 and 15r^3, which are like terms because they both contain r^3. We combined them by adding their coefficients (3 + 15) to get 18r^3, simplifying the expression. This step is essential before taking the derivative, as it makes the application of the power rule straightforward and the calculation less cluttered.

Why it's crucial:

Imagine you're organizing a collection of books by series. Combining like terms is similar to stacking books from the same series together. Just as this makes it easier to count how many books you have in each series, combining like terms in an algebraic expression makes it clearer and easier to operate on mathematically.