Question

How do you differentiate $f(x)=\dfrac{4}{\cos x}$ ?

Answer 1

Hint: We have to differentiate the given expression and we will be using the quotient rule. The given expression is already in the fraction so we need not do any changes to the expression. We will directly use the formula of the quotient rule and substitute the appropriate values in the formula and solve it. Hence, we have the derivative of the given expression.

Complete step-by-step solution:
According to the given question, we are given expressions to differentiate. Since no specific method is prescribed, we will use quotient rules.
Quotient rule involves finding the differentiation of an expression with a differentiable numerator and denominator.
The expression we have is,
$f(x)=\dfrac{4}{\cos x}$-----(1)
The given expression is already in a fraction form with differentiable numerator and denominator. So we will begin with writing the formula of quotient rule. We have,
\[\dfrac{d}{dx}\left( \dfrac{u}{v} \right)=\dfrac{v\dfrac{d}{dx}(u)-u\dfrac{d}{dx}(v)}{{{v}^{2}}}\]-----(2)
We will start substituting the required values in the above formula and we get,
\[\Rightarrow f'(x)=\dfrac{\cos x\dfrac{d}{dx}(4)-4\dfrac{d}{dx}(\cos x)}{{{\cos }^{2}}x}\]-----(3)
We further solve the equation (3), we get,
\[\Rightarrow f'(x)=\dfrac{\cos x(0)-4(-\sin x)}{{{\cos }^{2}}x}\]
As we know that derivative of a constant is zero, that is, \[\dfrac{d}{dx}(4)=0\] and the derivative of the cosine function is negative sine function, that is, \[\dfrac{d}{dx}(\cos x)=-\sin x\].
We get,
\[\Rightarrow f'(x)=\dfrac{4\sin x}{{{\cos }^{2}}x}\]-----(4)
We can write the equation (4), in a compact form as well. We got,
\[\Rightarrow f'(x)=\dfrac{4\sin x}{\cos x}.\dfrac{1}{\cos x}\]-----(5)
We know that, \[\dfrac{\sin x}{\cos x}=\tan x\] and \[\dfrac{1}{\cos x}=\sec x\], applying this in equation (5), we get,
\[\Rightarrow f'(x)=4\tan x\sec x\]
Therefore, the differentiation of the given expression is \[f'(x)=4\tan x\sec x\].

Note: We chose to differentiate the given expression using quotient rule as the expression was given in the fraction form and the numerator and denominator being differentiable. Thus, the conditions for quotients had been met, so we used this rule. We could also use chain rule to differentiate the given expression and we will still get the same result.