Find the derivative of [tex]y=\ln \left(x^3\right)+\sin x[/tex].
Answer (2)
Simplify the logarithmic term: ln ( x 3 ) = 3 ln ( x ) .
Differentiate 3 ln ( x ) to get x 3 .
Differentiate sin x to get cos x .
Combine the derivatives: d x d y = x 3 + cos x .
x 3 + cos x
Explanation
Problem Analysis We are given the function y = ln ( x 3 ) + sin x and we need to find its derivative with respect to x .
Simplifying the Function First, we can simplify the logarithmic term using the power rule for logarithms: ln ( x 3 ) = 3 ln ( x ) . So, our function becomes y = 3 ln ( x ) + sin x .
Differentiating the Logarithmic Term Now, we find the derivative of each term separately. The derivative of 3 ln ( x ) with respect to x is 3 ⋅ x 1 = x 3 .
Differentiating the Trigonometric Term The derivative of sin x with respect to x is cos x .
Combining the Derivatives Finally, we add the derivatives of the two terms to find the derivative of y : d x d y = x 3 + cos x
Final Answer Therefore, the derivative of y = ln ( x 3 ) + sin x is d x d y = x 3 + cos x .
Examples
In physics, if x represents time and y represents the position of an object, then the derivative d x d y gives the object's velocity. For example, if the position of an object is given by y = ln ( t 3 ) + sin t , the velocity of the object at any time t is given by t 3 + cos t . This allows us to understand how the object's velocity changes over time.
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