Question

for the following demand equation, differentiate implicitly to find dp/dx

p^{4}+p - 6x = 65

\frac{dp}{dx}=\square

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:

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