tutorial exercise use logarithmic differentiation to find the derivative of the function. y = x^3 cos(x)…

tutorial exercise use logarithmic differentiation to find the derivative of the function. y = x^3 cos(x) step 1 using properties of logarithms, we can rewrite the equation as ln(y)=ln(x^3 cos(x)), which is equivalent to ln(y)=(3 cos(x))(ln(x)). step 2 note that 3 cos(x) ln(x) is a product. therefore, its derivative is given by (3 cos(x))( )+(ln(x))( ).

tutorial exercise use logarithmic differentiation to find the derivative of the function. y = x^3 cos(x) step 1 using properties of logarithms, we can rewrite the equation as ln(y)=ln(x^3 cos(x)), which is equivalent to ln(y)=(3 cos(x))(ln(x)). step 2 note that 3 cos(x) ln(x) is a product. therefore, its derivative is given by (3 cos(x))( )+(ln(x))( ).

Answer

Explanation:

Step 1: Apply logarithm property

Given (y = x^{3\cos(x)}), taking the natural - logarithm of both sides, we use the property (\ln(a^{b})=b\ln(a)). So, (\ln(y)=\ln(x^{3\cos(x)}) = 3\cos(x)\ln(x)).

Step 2: Differentiate using product rule

The product rule states that if (u = 3\cos(x)) and (v=\ln(x)), and (y = uv), then (y^\prime=u^\prime v + uv^\prime). First, find (u^\prime): The derivative of (u = 3\cos(x)) with respect to (x) is (u^\prime=- 3\sin(x)) (using the derivative formula (\frac{d}{dx}(\cos(x))=-\sin(x))). Second, find (v^\prime): The derivative of (v = \ln(x)) with respect to (x) is (v^\prime=\frac{1}{x}). Then, (\frac{d}{dx}(3\cos(x)\ln(x))=-3\sin(x)\ln(x)+\frac{3\cos(x)}{x}). Since (\frac{d}{dx}(\ln(y))=\frac{y^\prime}{y}), and (\frac{d}{dx}(\ln(y))=\frac{d}{dx}(3\cos(x)\ln(x))=-3\sin(x)\ln(x)+\frac{3\cos(x)}{x}), we can solve for (y^\prime): (y^\prime=y\left(-3\sin(x)\ln(x)+\frac{3\cos(x)}{x}\right)=x^{3\cos(x)}\left(-3\sin(x)\ln(x)+\frac{3\cos(x)}{x}\right))

Answer:

(x^{3\cos(x)}\left(-3\sin(x)\ln(x)+\frac{3\cos(x)}{x}\right))