88. (a) Write | x | − sx 2 and use the Chain Rule to show that
d
dx | x | − x| x |
(b) If fsxd − | sin x |, fnd f9sxd and sketch the graphs of f
and f9. Where is f not differentiable?
(c) If tsxd − sin | x |, fnd t9sxd and sketch the graphs of t
and t9. Where is t not differentiable?
(a) To find the derivative of |x| - √(x^2), we can rewrite it as:
f(x) = |x| - √(x^2)
To differentiate this using the chain rule, we need to consider the derivative of the absolute value function and the square root function separately.
For the absolute value function, we know that its derivative with respect to x is:
d/dx |x| = x/|x|
And for the square root function, its derivative is:
d/dx √(x^2) = (1/2) * (2x) / sqrt(x^2) = x / √(x^2) = x / |x|
Now, we can apply the chain rule by multiplying the derivatives of the inner and outer functions:
d/dx |x| - √(x^2) = (x/|x|) - (x/|x|) = 0
Therefore, the derivative of |x| - √(x^2) is zero.
(b) Given fs(x) = -|sin(x)|, we can find f'(9) by applying the chain rule. Let's start by finding the derivative of the absolute value of sin(x):
d/dx |sin(x)| = cos(x) * d/dx(sin(x))
Using the chain rule, we have:
d/dx |sin(x)| = cos(x) * cos(x) = cos^2(x)
Now, we can find f'(x) by multiplying the derivative of |sin(x)| with -1:
f'(x) = -cos^2(x)
To find f'(9), we substitute x with 9:
f'(9) = -cos^2(9)
To sketch the graph of f(x) and f'(x), we need more information.
(c) Given ts(x) = -sin(|x|), to find t'(x), we can differentiate the function using the chain rule. Let's start with the derivative of the absolute value of x:
d/dx |x| = x / |x|
Now, let's find the derivative of -sin(|x|):
d/dx -sin(|x|) = -cos(|x|) * d/dx(|x|)
Using the chain rule, we have:
d/dx -sin(|x|) = -cos(|x|) * (x / |x|) = -x * cos(|x|) / |x|
Therefore, t'(x) = -x * cos(|x|) / |x|
To sketch the graphs of t(x) and t'(x), we need more information.
As for where t(x) and f(x) are not differentiable, it occurs where the derivative does not exist or there is a discontinuity in the function. In both cases, it would be necessary to analyze the behavior and particularities of the functions to determine where they are not differentiable.