Pregunta
Differentiate the function \( f(x)=10 \sin x-3 \cos x \)
Ask by Marshall Brooks. in the United States
Feb 27,2025
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The derivative of \( f(x) = 10 \sin x - 3 \cos x \) is \( f'(x) = 10 \cos x + 3 \sin x \).
Solución
To differentiate f(x) = 10 sin x - 3 cos x, we apply the derivatives of the sine and cosine functions. The derivative of sin x is cos x, and the derivative of cos x is -sin x.
Step-by-step:
1. Differentiate 10 sin x:
The derivative of 10 sin x is 10 cos x.
2. Differentiate -3 cos x:
The derivative of -3 cos x is -3(-sin x) = 3 sin x.
Combining these results gives:
f'(x) = 10 cos x + 3 sin x.
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Simplifique esta solución Extra Insights
The derivative of the function \( f(x) = 10 \sin x - 3 \cos x \) is \( f'(x) = 10 \cos x + 3 \sin x \). This means that the rate of change of the function \( f(x) \) with respect to \( x \) combines the effects of both sine and cosine, showcasing a dynamic relationship between the two functions. For those eager to explore further, trigonometric functions pop up in various delightful settings! From modeling tides, sound waves, and even the motion of pendulums, the interplay of sine and cosine functions is a key player. Delve into how these mathematical concepts tie into real-world phenomena and you'll discover the harmony in their oscillations!
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