if ( f(x)=\frac{3 x^{2} \tan x}{sec x} ), find ( f^{prime}(x)= ) find ( f^{prime}(1)= )

Answer:

$f^{\prime}(x)=6x\sin x + 3x^{2}\cos x$; $f^{\prime}(1)=6\sin1 + 3\cos1$

Explanation:

Step1: Simplify the function

Since $\frac{\tan x}{\sec x}=\sin x$, then $f(x)=\frac{3x^{2}\tan x}{\sec x}=3x^{2}\sin x$.

Step2: Apply the product rule

The product rule is $(uv)^\prime = u^\prime v+uv^\prime$, where $u = 3x^{2}$ and $v=\sin x$. First, find $u^\prime$: $u^\prime=(3x^{2})^\prime = 6x$. Second, find $v^\prime$: $v^\prime = (\sin x)^\prime=\cos x$. Then $f^\prime(x)=(3x^{2})^\prime\sin x+3x^{2}(\sin x)^\prime$.

Step3: Calculate $f^\prime(x)$

Substitute $u^\prime$ and $v^\prime$ into the product - rule formula: $f^\prime(x)=6x\sin x + 3x^{2}\cos x$.

Step4: Calculate $f^\prime(1)$

Substitute $x = 1$ into $f^\prime(x)$: $f^\prime(1)=6\times1\times\sin1+3\times1^{2}\times\cos1=6\sin1 + 3\cos1$.