Question
\( y=\cot (5 \cdot x+3) \) Find \( \frac{d y}{d x} \) \( \frac{d y}{d x}= \)
Ask by Estrada Stanley. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(\frac{dy}{dx} = -5 \csc^2(5x + 3)\)
Solution
Given y = cot(5x + 3), we need to differentiate using the chain rule. Recall that the derivative of cot(u) with respect to u is -csc²(u), and then multiply by the derivative of the inner function u = 5x + 3.
Step 1: Differentiate the outer function:
d/dx [cot(u)] = -csc²(u) · (du/dx)
Step 2: Differentiate u = 5x + 3 with respect to x:
du/dx = 5
Thus, combining these results:
dy/dx = -csc²(5x + 3) · 5
Final answer:
dy/dx = -5 csc²(5x + 3)
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Simplify this solution Mind Expander
To find the derivative \( \frac{d y}{d x} \) of the function \( y = \cot(5x + 3) \), we use the chain rule. The derivative of \( \cot(u) \) is \( -\csc^2(u) \) and we multiply by the derivative of the inner function \( u = 5x + 3 \) which is 5. So, \[ \frac{d y}{d x} = -\csc^2(5x + 3) \cdot \frac{d}{dx}(5x + 3) = -\csc^2(5x + 3) \cdot 5 \] Thus, \[ \frac{d y}{d x} = -5 \csc^2(5x + 3) \]
