If w = arcsin(x - 3), z = cos(ln(x)) then find d/dz (w)
1/(sin(ln(x)) * sqrt(1 + (x + 3) ^ 2))
- 1/(sin(ln(x)) * sqrt(1 - (x - 3)))
1/(sin(ln(x)) * sqrt(1 - (x - 3) ^ 2))
- 1/(sin(ln(x)) * sqrt(1 - (x - 3) ^ 2))
You asked:
Given \( w = \arcsin\left( x - 3 \right) \), \( z = \cos\left( \ln\left( x \right) \right) \), evaluate \( \frac{d}{dz}\left[w\right] \), \( \frac{1}{\sin\left( \ln\left( x \right) \right) \cdot \sqrt{1 + {\left( x + 3 \right)}^{2}}} \), \( \frac{-1}{\sin\left( \ln\left( x \right) \right) \cdot \sqrt{1 - \left(x - 3\right)}} \), \( \frac{1}{\sin\left( \ln\left( x \right) \right) \cdot \sqrt{1 - {\left( x - 3 \right)}^{2}}} \), \( \frac{-1}{\sin\left( \ln\left( x \right) \right) \cdot \sqrt{1 - {\left( x - 3 \right)}^{2}}} \).
MathBot Answer:
The system of equations has infinitely many solutions.
$$\begin{aligned}\frac{d}{d z} w&=0\\\frac{1}{\sqrt{\left(x + 3\right)^{2} + 1} \sin{\left(\log{\left(x \right)} \right)}}&=\frac{1}{\sqrt{\left(\sin{\left(w \right)} + 6\right)^{2} + 1} \sin{\left(\log{\left(\sin{\left(w \right)} + 3 \right)} \right)}}\\- \frac{1}{\sqrt{1 - \left(x - 3\right)} \sin{\left(\log{\left(x \right)} \right)}}&=- \frac{1}{\sqrt{1 - \sin{\left(w \right)}} \sin{\left(\log{\left(\sin{\left(w \right)} + 3 \right)} \right)}}\\\frac{1}{\sqrt{1 - \left(x - 3\right)^{2}} \sin{\left(\log{\left(x \right)} \right)}}&=\frac{1}{\sqrt{1 - \sin^{2}{\left(w \right)}} \sin{\left(\log{\left(\sin{\left(w \right)} + 3 \right)} \right)}}\\- \frac{1}{\sqrt{1 - \left(x - 3\right)^{2}} \sin{\left(\log{\left(x \right)} \right)}}&=- \frac{1}{\sqrt{1 - \sin^{2}{\left(w \right)}} \sin{\left(\log{\left(\sin{\left(w \right)} + 3 \right)} \right)}}\end{aligned}$$