Use the table below to evaluate the derivative with respect to x of g of f of 2 times x at x = 1.
x 1 2 3 4
f(x) 6 1 8 2
f ′(x) 6 1 8 2
g(x) 1 4 4 3
g ′(x) 9 5 5 –4
That would be
f'(g(2x))g'(2x)(2x)'
= f'(g(2))g'(2)(2)
= f'(4)g'(2)(2)
= 2*4*2
= 16
Well, let's break it down step by step.
First, we need to find f(2x). Since we are given the values of f(x), we can substitute 2x into the table:
f(2x) = f(2*1) = f(2) = 1
Next, we need to find g(f(2x)). Using the values of g(x), we substitute f(2x) into the table:
g(f(2x)) = g(1) = 4
Finally, we need to find the derivative of g(f(2x)) with respect to x. Using the values of g'(x), we substitute f(2x) into the table:
g'(f(2x)) = g'(1) = 5
So, the derivative with respect to x of g(f(2x)) at x = 1 is 5. However, I must confess, the table makes me feel a bit like a circus act juggling all these values. I hope my performance hasn't been too clownish!
To evaluate the derivative with respect to x of g(f(2x)) at x = 1, we need to follow these steps:
Step 1: Evaluate f(2x) at x = 1
f(2 * 1) = f(2) = 1
Step 2: Evaluate g(f(2x)) using the result from step 1
g(f(2x)) = g(1)
Step 3: Evaluate g(1) using the given values in the table
g(1) = 4
Therefore, the derivative with respect to x of g(f(2x)) at x = 1 is 4.
To evaluate the derivative with respect to x of g of f of 2 times x at x = 1, we need to use the chain rule. The chain rule states that the derivative of composition of two functions is calculated by multiplying the derivative of the outer function with the derivative of the inner function.
First, let's evaluate f of 2 times x at x = 1. To do this, we substitute x = 1 into the expression for f(x):
f(2x) = f(2 * 1) = f(2) = 1
Next, we need to find the derivative of f(2x) with respect to x. From the given table, we can see that the derivative of f(x) is the same as the function itself:
f ′(x) = 6, 1, 8, 2
So, the derivative of f(2x) is also:
f ′(2x) = 6, 1, 8, 2
Now, we can evaluate g of f of 2 times x at x = 1. To do this, we substitute the result of f(2x) at x = 1 into the expression for g(x):
g(f(2x)) = g(1)
Finally, we need to find the derivative of g(f(2x)) with respect to x. From the given table, we can see that the derivative of g(x) is given by:
g ′(x) = 9, 5, 5, -4
Since g(1) does not depend on x, its derivative with respect to x will be 0.
Therefore, the derivative with respect to x of g of f of 2 times x at x = 1 is 0.