To find the derivative of each given function, let's go through the calculations step-by-step.
### 1. Finding the derivative of [tex]\( y = \cos^{-1}\left(\frac{7}{x}\right) \)[/tex]:
1. Let [tex]\( y = \cos^{-1}\left(\frac{7}{x}\right) \)[/tex].
2. Recall that [tex]\( \frac{d}{dx} \cos^{-1}(u) = -\frac{1}{\sqrt{1 - u^2}} \)[/tex] where [tex]\( u \)[/tex] is a function of [tex]\( x \)[/tex].
3. Here, [tex]\( u = \frac{7}{x} \)[/tex].
4. First, find [tex]\( \frac{du}{dx} \)[/tex].
[tex]\[
u = \frac{7}{x} \implies \frac{du}{dx} = -\frac{7}{x^2}
\][/tex]
5. Now, apply the chain rule:
[tex]\[
\frac{dy}{dx} = -\frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx}
\][/tex]
6. Substitute [tex]\( u = \frac{7}{x} \)[/tex] and [tex]\( \frac{du}{dx} = -\frac{7}{x^2} \)[/tex]:
[tex]\[
\frac{dy}{dx} = -\frac{1}{\sqrt{1 - \left(\frac{7}{x}\right)^2}} \cdot \left(-\frac{7}{x^2}\right)
\][/tex]
7. Simplify:
[tex]\[
\frac{dy}{dx} = \frac{7}{x^2 \cdot \sqrt{1 - \frac{49}{x^2}}}
\][/tex]
8. Further simplification gives:
[tex]\[
\frac{dy}{dx} = \frac{7}{x^2 \cdot \sqrt{\frac{x^2 - 49}{x^2}}}
\][/tex]
9. Simplifying inside the square root:
[tex]\[
\frac{dy}{dx} = \frac{7}{x^2 \cdot \frac{\sqrt{x^2 - 49}}{x}}
\][/tex]
10. Finally:
[tex]\[
\frac{dy}{dx} = \frac{7}{x^2} \cdot \frac{x}{\sqrt{x^2 - 49}} = \frac{7}{x \cdot \sqrt{x^2 - 49}}
\][/tex]
### 2. Finding the derivative of [tex]\( y = \sin^{-1}(4 - x) \)[/tex]:
1. Let [tex]\( y = \sin^{-1}(4 - x) \)[/tex].
2. Recall that [tex]\( \frac{d}{dx} \sin^{-1}(u) = \frac{1}{\sqrt{1 - u^2}} \)[/tex] where [tex]\( u \)[/tex] is a function of [tex]\( x \)[/tex].
3. Here, [tex]\( u = 4 - x \)[/tex].
4. First, find [tex]\( \frac{du}{dx} \)[/tex].
[tex]\[
u = 4 - x \implies \frac{du}{dx} = -1
\][/tex]
5. Now, apply the chain rule:
[tex]\[
\frac{dy}{dx} = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx}
\][/tex]
6. Substitute [tex]\( u = 4 - x \)[/tex] and [tex]\( \frac{du}{dx} = -1 \)[/tex]:
[tex]\[
\frac{dy}{dx} = \frac{1}{\sqrt{1 - (4 - x)^2}} \cdot (-1)
\][/tex]
7. Simplify:
[tex]\[
\frac{dy}{dx} = -\frac{1}{\sqrt{1 - (4 - x)^2}}
\][/tex]
These calculations yield the following derivatives:
- The derivative of [tex]\( y = \cos^{-1}\left(\frac{7}{x}\right) \)[/tex] is [tex]\( \frac{7}{x \cdot \sqrt{x^2 - 49}} \)[/tex].
- The derivative of [tex]\( y = \sin^{-1}(4 - x) \)[/tex] is [tex]\( -\frac{1}{\sqrt{1 - (4 - x)^2}} \)[/tex].
