Question

find out the value of derivative of log((10 cos x)*(5 sin x)) at x=pi/4
A.) 1
B.) infinity
C.) -1
D.) 0

Answer 1

Answer:

A.) 1

Step-by-step explanation:

In mathematics, a derivative is the rate at which a function changes in relation to a variable. Calculus and differential equations issues must be solved using derivatives. It is a crucial idea that is incredibly helpful in a variety of contexts: in daily life, the derivative may inform you how fast you are driving or assist you in predicting stock market changes; in machine learning, derivatives are crucial for function optimization.

The usage of a logarithm can be utilised to address issues that cannot be resolved using the notion of exponents alone. A logarithm is just another way to describe exponents. Log interpretation is not that tough. It suffices to know that an exponential equation may also be written as a logarithmic equation in order to comprehend logarithms.

$\begin{aligned}& \log _{10}\left(\left(10 \cos \left(\frac{\pi}{4}\right)\right)\left(5 \sin \left(\frac{\pi}{4}\right)\right)\right) \\& =\log _{10}\left(10 \cos \left(\frac{\pi}{4}\right) \cdot 5 \sin \left(\frac{\pi}{4}\right)\right)\end{aligned}$

Use the following trivial identity: $\cos \left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$

Use the following trivial identity: $\sin \left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$

$=\log _{10}\left(10 \cdot \frac{\sqrt{2}}{2} \cdot 5 \cdot \frac{\sqrt{2}}{2}\right)$

Simplify $\log _{10}\left(10 \cdot \frac{\sqrt{2}}{2} \cdot 5 \cdot \frac{\sqrt{2}}{2}\right): 2 \log _{10}(5)$

$=2 \log _{10}(5)$= 1.39794

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