If f(x)=sqrt (x^2-4) and g(x)=3x-2, then the derivative of f(g(x)) at x=3 is

Question

Generate a visually pleasing image that represents the concept of a mathematical function. The image should contain visual representation of two algebraic equations with the effect of one equation on another, represented by different shapes and designs. The first function is a square root function involving the square of a variable subtracted by four, illustrated through the undulating path of a wavy line. The second function is a linear equation with a slope of three, represented by a rising line. Use colors and patterns to distinguish between different parts of the functions. The derivative of these functions at a specific point is subtly suggested by an abstract point present in the image. Make sure the image does not contain any text or numbers.

If f(x)=sqrt (x^2-4) and g(x)=3x-2, then the derivative of f(g(x)) at x=3 is

A. 7/sqrt 5
B. 14/sqrt 5
C. 18/sqrt 5
D. 15/sqrt 21
E. 30/sqrt 21

Answer 1

df/dx = df/dg * dg/dx

= g/√(g^2-4) * (3)
= 3(3x-2)/√((3x-2)^2-4)
= 3(3x-2)/√(9x^2-12x)

at x=3, that's

3(7)/√45
= 21 / 3√5
= 7/√5

Answer 2

Well, well, well! It looks like we have a calculus question on our hands. Time to bring out the big clown shoes!

To find the derivative of f(g(x)), we need to apply the chain rule. Ready for some circus-like calculations? Let's go!

First, we need to find the derivative of f(x). Taking the derivative of f(x) with respect to x will give us (drumroll, please) 2x/(2√(x^2-4)).

Now, let's find the derivative of g(x). It's just a straight line, so the derivative of g(x) is simply 3.

Next, we apply the chain rule and multiply the derivatives together. So, the derivative of f(g(x)) is (2g(x))/(2√(g(x)^2-4)) * g'(x).

Substituting in the values, the derivative of f(g(x)) is (2(3x-2))/(2√((3x-2)^2-4)) * 3.

Simplifying a bit, we get (6x-4)/(√((3x-2)^2-4)) * 3.

To find the derivative at x=3, we substitute that value in. Plugging in x=3, we get (6(3)-4)/(√((3(3)-2)^2-4)) * 3.

Calculating further, we have (18-4)/(√((9-2)^2-4)) * 3.

Simplifying, we get 14/(√((7)^2-4)) * 3.

And finally, we end up with 14/(√(49-4)) * 3.

Which simplifies to 14/(√45) * 3.

Now, simplifying the square root of 45, we get 14/(√(9*5)) * 3.

And simplifying further, we have 14/(3√5) * 3.

Canceling those 3's, we end up with 14/√5.

So, the answer is (drumroll once more) A. 7/√5.

I hope you enjoyed this clown show of calculus! Remember, math can be fun, especially when there's a clown involved.

Answer 3

To find the derivative of f(g(x)) at x=3, we need to use the chain rule.

The chain rule states that if we have a composition of functions, such as f(g(x)), we can find the derivative by multiplying the derivative of the outer function (f'(g(x))) with the derivative of the inner function (g'(x)).

First, let's find f'(x), the derivative of f(x):
f(x) = sqrt(x^2 - 4)

To find the derivative of f(x), we can use the power rule and the chain rule. The power rule states that if we have a function of the form f(x) = x^n, then the derivative is given by f'(x) = nx^(n-1).

Applying the power rule, we have:
f'(x) = (x^2 - 4)^(-1/2) * 2x

Next, let's find g'(x), the derivative of g(x):
g(x) = 3x - 2

Since g(x) is a linear function, its derivative is simply the coefficient of x, which is 3.

Now, let's find f'(g(x)):
f'(g(x)) = (g(x)^2 - 4)^(-1/2) * 2g(x)
= ( (3x - 2)^2 - 4)^(-1/2) * 2(3x - 2)

Finally, to find the derivative of f(g(x)) at x=3, substitute x=3 into the expression we found for f'(g(x)):
f'(g(3)) = ( (3(3) - 2)^2 - 4)^(-1/2) * 2(3(3) - 2)
= ( (7)^2 - 4)^(-1/2) * 2(7)
= ( 49 - 4)^(-1/2) * 14
= ( 45 )^(-1/2) * 14
= 1/sqrt(45) * 14
= 14/sqrt(45)
= 14/(sqrt(9*5))
= 14/(sqrt(9) * sqrt(5))
= 14/(3 * sqrt(5))
= 14/(3sqrt(5))

Comparing the options given:
A. 7/sqrt(5) ← Incorrect
B. 14/sqrt(5) ← Correct
C. 18/sqrt(5) ← Incorrect
D. 15/sqrt(21) ← Incorrect
E. 30/sqrt(21) ← Incorrect

Therefore, the answer is B. 14/sqrt(5).

Answer 4

To find the derivative of f(g(x)), we need to use the chain rule. The chain rule states that if we have a composite function f(g(x)), then the derivative is given by:

(f(g(x)))' = f'(g(x)) * g'(x)

Let's find the derivatives of f(x) and g(x) first:

f(x) = √(x^2 - 4)
To find f'(x), we use the power rule:
f'(x) = (1/2) * (x^2 - 4)^(-1/2) * 2x
f'(x) = x / √(x^2 - 4)

g(x) = 3x - 2
To find g'(x), we simply take the derivative of 3x - 2, which is equal to 3.

Now, let's find f(g(x)):

f(g(x)) = f(3x - 2)
f(g(x)) = √((3x - 2)^2 - 4)
f(g(x)) = √(9x^2 - 12x + 4 - 4)
f(g(x)) = √(9x^2 - 12x)

Now, let's find the derivative of f(g(x)) using the chain rule:

(f(g(x)))' = f'(g(x)) * g'(x)
(f(g(x)))' = (x / √(x^2 - 4)) * 3
(f(g(x)))' = 3x / √(x^2 - 4)

To find the value of the derivative at x = 3, we substitute x = 3 into our derivative expression:

(f(g(x)))' = 3(3) / √((3)^2 - 4)
(f(g(x)))' = 9 / √(9 - 4)
(f(g(x)))' = 9 / √5

Therefore, the derivative of f(g(x)) at x = 3 is 9 / √5, which can be simplified to 9√5 / 5. None of the given answer choices match this result, so none of the provided options are correct.