Suppose that functions f and g and their derivatives have the following values at x=2 and x=3.

Suppose that functions f and g and their derivatives have...

Key Concepts

Power Rule

The power rule simplifies the process of differentiating functions raised to a power. It states that if a function is of the form x^n, its derivative is n times x raised to the (n-1) power. This rule is often combined with the chain rule when the base of the exponent is itself a function of x.

Chain Rule

The chain rule is essential for differentiating composite functions, where one function is nested inside another. It dictates that the derivative of the composite function is the derivative of the outer function (evaluated at the inner function) multiplied by the derivative of the inner function.

Quotient Rule

The quotient rule provides a method for differentiating a quotient of two functions. It involves taking the derivative of the numerator multiplied by the denominator, subtracting the numerator multiplied by the derivative of the denominator, and then dividing the entire expression by the square of the denominator.

Constant Multiple Rule

This rule states that when a function is multiplied by a constant, its derivative is the constant multiplied by the derivative of the function. It simplifies the differentiation process when scaling a function by a fixed number.

Sum/Difference Rule

The sum/difference rule indicates that the derivative of the sum (or difference) of functions is simply the sum (or difference) of their derivatives. This allows for the independent differentiation of each term in an expression.

Product Rule

The product rule is used when differentiating the product of two functions. It states that the derivative of the product is equal to the first function times the derivative of the second, plus the second function times the derivative of the first. This rule is crucial when the functions are multiplied together.

Transcript

00:01 Section 3 .6, problem number 89, they're just going to drill us to see if we are good with chain rule and derivatives.

00:08 Okay? so they're giving us functions f and g, not telling us what those functions are, but asking for different derivatives of different combinations and functions built on those.

00:18 So how do i find the derivative of 2f prime of x? excuse me, 2f of x.

00:22 How do i find that derivative and evaluated x equal 2? well, the derivative of this, this is going to be 2f prime of x.

00:31 And i just need to evaluate this at x equal to 2.

00:35 So that's going to be 2 times f prime of 2.

00:41 And what is f prime of 2? f prime of 2 is 1 3rd.

00:46 So this is 2 times 1 3rd, which is 2 thirds.

00:52 So that is part a to this particular problem, which is 2 thirds.

00:57 Now let's move on to part b.

00:59 So in part b they're asking us to find the derivative of f of x plus g of x so find that derivative evaluated at x equal three with a derivative of the sum this is just going to be f prime of x plus g prime of x all of this evaluated when x is equal to three so this is just f prime of three of 3.

01:37 So what is f prime of 3? f prime of 3 is 2 pi.

01:43 G prime of 3, g prime of 3 is 5.

01:51 And so that is my answer.

01:53 2 pi plus 5 is the answer of the derivative of f plus g evaluated at x equal 3.

02:01 Part c, we're asked to find the derivative of f of x times g of x.

02:13 And to evaluate the this at x equal three.

02:18 Well, this is the product rule.

02:20 So the derivative is going to be f prime of x times g of x plus g prime of x times f of x.

02:32 And i need to evaluate this when x is equal to 3.

02:36 So this turns into f prime of 3, g of 3, plus g prime of 3, f prime of 3, f prime of 3, f of 3.

02:49 So f prime of 3 is 2 pi, g of 3, so g of 3 is negative 4, plus g prime of 3, g prime of 3 is 5, and then, so g prime of 3 times f of 3, f of 3 is 3.

03:15 So this becomes 15 minus 8 pi.

03:24 8 pi.

03:29 So moving on, part d.

03:32 So in part d, we're asked to find the derivative of f of x divided by g of x.

03:42 So find that derivative when x is equal to 2.

03:47 So this is the quotient rule.

03:49 So this derivative is going to be, so it's going to be f prime of x, g of x, minus g prime of x, f of x over g of x squared so to evaluate this and so in part d we're evaluating this when x is equal to two is that right so evaluate this when x is equal to two so what do we get here f prime of x okay when x is equal to two is one -third so this is one -third g of x so g of two g of two is two minus g prime of two is negative three and then f of two is eight over g of two squared so g of two is two so over two of two is two so over two squared so over two squared so this becomes what two thirds plus 24, all of this divided by four.

05:29 And so what is this? 24 plus two thirds.

05:34 So this can be another way of thinking of this.

05:38 Let's just see.

05:39 This is 24 over 4 plus 1 fourth of two thirds.

05:47 So that's six plus 212.

05:51 Which is one six, so this is six and one six is the final answer there.

06:03 Moving on, part e of this question.

06:07 So they want us to find the derivative of f of g of x when x is equal to two.

06:15 This is just the definition of the chain rule.

06:18 So this is going to be f prime of g of x times g prime of x.

06:25 So substituting x equal to two, this becomes f prime of g of 2 times g prime of 2.

06:38 So this is f prime...