Derivative of (4-sec(x))/(tan(x))
The Correct Answer and Explanation is:
To find the derivative of the function 4−sec(x)tan(x)\frac{4 – \sec(x)}{\tan(x)}tan(x)4−sec(x), we’ll use the quotient rule, which is applied to functions of the form u(x)v(x)\frac{u(x)}{v(x)}v(x)u(x).
The quotient rule states: ddx(u(x)v(x))=v(x)⋅u′(x)−u(x)⋅v′(x)(v(x))2\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) – u(x) \cdot v'(x)}{(v(x))^2}dxd(v(x)u(x))=(v(x))2v(x)⋅u′(x)−u(x)⋅v′(x)
Where u(x)=4−sec(x)u(x) = 4 – \sec(x)u(x)=4−sec(x) and v(x)=tan(x)v(x) = \tan(x)v(x)=tan(x).
Step 1: Differentiate u(x)=4−sec(x)u(x) = 4 – \sec(x)u(x)=4−sec(x)
The derivative of a constant is zero, so the derivative of 4 is 0. To differentiate −sec(x)-\sec(x)−sec(x), we use the chain rule: ddxsec(x)=sec(x)tan(x)\frac{d}{dx} \sec(x) = \sec(x) \tan(x)dxdsec(x)=sec(x)tan(x)
Thus, the derivative of u(x)u(x)u(x) is: u′(x)=−sec(x)tan(x)u'(x) = -\sec(x) \tan(x)u′(x)=−sec(x)tan(x)
Step 2: Differentiate v(x)=tan(x)v(x) = \tan(x)v(x)=tan(x)
The derivative of tan(x)\tan(x)tan(x) is sec2(x)\sec^2(x)sec2(x), so: v′(x)=sec2(x)v'(x) = \sec^2(x)v′(x)=sec2(x)
Step 3: Apply the quotient rule
Now we apply the quotient rule: ddx(4−sec(x)tan(x))=tan(x)⋅(−sec(x)tan(x))−(4−sec(x))⋅sec2(x)tan2(x)\frac{d}{dx} \left( \frac{4 – \sec(x)}{\tan(x)} \right) = \frac{\tan(x) \cdot (-\sec(x) \tan(x)) – (4 – \sec(x)) \cdot \sec^2(x)}{\tan^2(x)}dxd(tan(x)4−sec(x))=tan2(x)tan(x)⋅(−sec(x)tan(x))−(4−sec(x))⋅sec2(x)
Step 4: Simplify the expression
We can simplify each term in the numerator: =−sec(x)tan2(x)−(4−sec(x))sec2(x)tan2(x)= \frac{-\sec(x) \tan^2(x) – (4 – \sec(x)) \sec^2(x)}{\tan^2(x)}=tan2(x)−sec(x)tan2(x)−(4−sec(x))sec2(x)
Distribute sec2(x)\sec^2(x)sec2(x) in the second term: =−sec(x)tan2(x)−4sec2(x)+sec3(x)tan2(x)= \frac{-\sec(x) \tan^2(x) – 4\sec^2(x) + \sec^3(x)}{\tan^2(x)}=tan2(x)−sec(x)tan2(x)−4sec2(x)+sec3(x)
Final answer:
The derivative of 4−sec(x)tan(x)\frac{4 – \sec(x)}{\tan(x)}tan(x)4−sec(x) is: −sec(x)tan2(x)−4sec2(x)+sec3(x)tan2(x)\frac{-\sec(x) \tan^2(x) – 4 \sec^2(x) + \sec^3(x)}{\tan^2(x)}tan2(x)−sec(x)tan2(x)−4sec2(x)+sec3(x)
This expression involves standard trigonometric derivatives and simplifications. It gives you the rate of change of the function 4−sec(x)tan(x)\frac{4 – \sec(x)}{\tan(x)}tan(x)4−sec(x) with respect to xxx.
