if ( f(x)=\frac{2 x^{2} \tan x}{sec x} ), find ( f^{prime}(x)= ) find ( f^{prime}(1)= )

Explanation:

Step1: Simplify the function

Use the trigonometric identity (\tan x=\frac{\sin x}{\cos x}) and (\sec x = \frac{1}{\cos x}). Then (f(x)=\frac{2x^{2}\tan x}{\sec x}=2x^{2}\sin x).

Step2: Apply the product rule

The product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = 2x^{2}), (u^\prime=4x) and (v=\sin x), (v^\prime=\cos x). So (f^\prime(x)=(2x^{2})^\prime\sin x+2x^{2}(\sin x)^\prime=4x\sin x + 2x^{2}\cos x).

Step3: Calculate (f^\prime(1))

Substitute (x = 1) into (f^\prime(x)). (f^\prime(1)=4\times1\times\sin(1)+2\times1^{2}\times\cos(1)=4\sin(1)+2\cos(1)).

Answer:

(f^\prime(x)=4x\sin x + 2x^{2}\cos x) (f^\prime(1)=4\sin(1)+2\cos(1))