Question

Differentiate implicitly to find \(d y / d x\). \(\cos x+\tan x y+5=0\)

Short answer

\(\frac{dy}{dx}=\frac{\sin x-\sec^2 x y }{\tan x}\)

Step-by-step solution

Differentiate each term with respect to \(x\)

When \(0\) is differentiated with respect to \(x\), it will result in \(0\). Differentiating \(\cos x\) with respect to \(x\) results in \(-\sin x\). To differentiate \(\tan x y\) with respect to \(x\), we apply the product rule and differentiate each variable within the term. First differentiate \(\tan x\) with respect to \(x\) while keeping \(y\) constant, and then differentiate \(y\) with respect to \(x\) while keeping \(\tan x\) constant. This process yields \(\sec^2 x y + \tan x \frac{dy}{dx}\). Lastly, differentiating the constant \(5\) with respect to \(x\) gives \(0\).

Write down the derivative after differentiation

After performing the differentiation, the resultant equation will be \(-\sin x + \sec^2 x y + \tan x \frac{dy}{dx} = 0\).

Solve for \(dy/dx\)

The next step is to solve the equation for \(\frac{dy}{dx}\). To do this, subtract \(-\sin x\) and \(\sec^2 x y\) from both sides of the equation to isolate \(\frac{dy}{dx}\) on one side. After performing the algebra, we get \(\frac{dy}{dx}=\frac{\sin x-\sec^2 x y }{\tan x}\).