4. calculate the following derivatives.\na. $\frac{d}{dx}(x^{2}\tan x + 5xsec x)$\nb. $\frac{d}{dw}left(\frac…

Explanation:

Step1: Apply sum - rule of derivatives

The sum - rule states that ((u + v)^\prime=u^\prime + v^\prime). For (y = x^{2}\tan x+5x\sec x), let (u = x^{2}\tan x) and (v = 5x\sec x). Then (\frac{d}{dx}(x^{2}\tan x + 5x\sec x)=\frac{d}{dx}(x^{2}\tan x)+\frac{d}{dx}(5x\sec x))

Step2: Apply product - rule for (\frac{d}{dx}(x^{2}\tan x))

The product - rule is ((uv)^\prime = u^\prime v+uv^\prime). Here (u = x^{2}), (u^\prime=2x), (v=\tan x), (v^\prime=\sec^{2}x). So (\frac{d}{dx}(x^{2}\tan x)=2x\tan x + x^{2}\sec^{2}x)

Step3: Apply product - rule for (\frac{d}{dx}(5x\sec x))

Here (u = 5x), (u^\prime=5), (v=\sec x), (v^\prime=\sec x\tan x). So (\frac{d}{dx}(5x\sec x)=5\sec x+5x\sec x\tan x)

Step4: Combine the results

(\frac{d}{dx}(x^{2}\tan x + 5x\sec x)=2x\tan x+x^{2}\sec^{2}x + 5\sec x+5x\sec x\tan x)

For part B:

Step1: Apply quotient - rule

The quotient - rule is ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}), where (u = 3w^{5/3}+5\cos w), (u^\prime = 5w^{2/3}-5\sin w), (v=\sin w + 2w^{3}), (v^\prime=\cos w+6w^{2}) (\frac{d}{dw}\left(\frac{3w^{5/3}+5\cos w}{\sin w + 2w^{3}}\right)=\frac{(5w^{2/3}-5\sin w)(\sin w + 2w^{3})-(3w^{5/3}+5\cos w)(\cos w + 6w^{2})}{(\sin w + 2w^{3})^{2}})

For part C:

Step1: Apply product - rule

Let (u=\sqrt{t}=t^{1/2}), (u^\prime=\frac{1}{2}t^{-1/2}), (v = \sec t), (v^\prime=\sec t\tan t) (\frac{d}{dt}(\sqrt{t}\cdot\sec t)=\frac{1}{2\sqrt{t}}\sec t+\sqrt{t}\sec t\tan t)

For part D:

Step1: First, rewrite (y = e^{\theta}\sin\theta\cos\theta=\frac{1}{2}e^{\theta}\sin(2\theta)) (using double - angle formula (\sin(2\theta)=2\sin\theta\cos\theta))

Step2: Apply product - rule

Let (u=\frac{1}{2}e^{\theta}), (u^\prime=\frac{1}{2}e^{\theta}), (v=\sin(2\theta)), (v^\prime = 2\cos(2\theta)) (\frac{d}{d\theta}(e^{\theta}\sin\theta\cos\theta)=\frac{1}{2}e^{\theta}\sin(2\theta)+e^{\theta}\cos(2\theta))

Answer:

A. (2x\tan x+x^{2}\sec^{2}x + 5\sec x+5x\sec x\tan x) B. (\frac{(5w^{2/3}-5\sin w)(\sin w + 2w^{3})-(3w^{5/3}+5\cos w)(\cos w + 6w^{2})}{(\sin w + 2w^{3})^{2}}) C. (\frac{1}{2\sqrt{t}}\sec t+\sqrt{t}\sec t\tan t) D. (\frac{1}{2}e^{\theta}\sin(2\theta)+e^{\theta}\cos(2\theta))