Question

7 use your calculator to sketch the graph of the derivative of the following functions. use the calculator’s graph to sketch your own graph below and then to identify the equation of the derivative function. (a) ( g(x)=sin x ) (b) ( f(x)=ln x ) (c) ( y = -ln|cos x| ) 8. for the following, state whether the function is continuous, differentiable, both, or neither at ( x = c ) a) b) c) d) d) e) f) g)

Explanation:

For Question 7:

Step1: Derive $g(x)=\sin x$

The derivative of $\sin x$ is a standard calculus result: $\frac{d}{dx}\sin x = \cos x$

Step2: Derive $f(x)=\ln x$

Using the derivative rule for natural log: $\frac{d}{dx}\ln x = \frac{1}{x}$ (for $x>0$)

Step3: Derive $y=-\ln|\cos x|$

Apply chain rule: Let $u=|\cos x|$, so $y=-\ln u$. $\frac{dy}{du}=-\frac{1}{u}$, $\frac{du}{dx}=-\frac{\cos x \cdot \sin x}{|\cos x|}$ (simplifies to $-\tan x$ for $|\cos x|
eq0$). Multiply: $\frac{dy}{dx}=-\frac{1}{|\cos x|} \cdot (-\tan x \cdot |\cos x|) = \tan x$

A function is continuous at $x=c$ if there is no break/hole at $x=c$, and differentiable if it is continuous and has no corner/cusp/vertical tangent at $x=c$.

• (a): The graph has a corner at $x=c$, so it is continuous but not differentiable.
• (b): The graph has a hole at $x=c$, so it is neither continuous nor differentiable.
• (c): The graph has a vertical asymptote at $x=c$, so it is neither continuous nor differentiable.
• (d): The graph has no break/corner at $x=c$, so it is both continuous and differentiable.

Answer:

(a) $g'(x)=\cos x$
(b) $f'(x)=\frac{1}{x}$ (for $x>0$)
(c) $y'=\tan x$ (for $x
eq \frac{\pi}{2}+k\pi, k\in\mathbb{Z}$)

---

For Question 8: