for the following demand equation, differentiate implicitly to find dp/dx\n\np^{4}+p - 6x =…

for the following demand equation, differentiate implicitly to find dp/dx\n\np^{4}+p - 6x = 65\n\n\\frac{dp}{dx}=\\square

for the following demand equation, differentiate implicitly to find dp/dx\n\np^{4}+p - 6x = 65\n\n\\frac{dp}{dx}=\\square

Answer

Explanation:

Step1: Differentiate each term with respect to (x)

Differentiate (p^{4}) using the chain - rule (\frac{d}{dx}(p^{4}) = 4p^{3}\frac{dp}{dx}), (\frac{d}{dx}(p)=\frac{dp}{dx}), (\frac{d}{dx}(-6x)=-6), and (\frac{d}{dx}(65) = 0). So the derivative of the entire equation (p^{4}+p - 6x=65) is (4p^{3}\frac{dp}{dx}+\frac{dp}{dx}-6 = 0).

Step2: Solve for (\frac{dp}{dx})

Factor out (\frac{dp}{dx}) from the left - hand side: (\frac{dp}{dx}(4p^{3}+1)=6). Then, divide both sides by ((4p^{3}+1)) to get (\frac{dp}{dx}=\frac{6}{4p^{3}+1}).

Answer:

(\frac{6}{4p^{3}+1})