Find the derivative and simplify your
answer.

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Mar 22,2025

Question

Find the derivative and simplify your answer. $$ A(x)=\int_{1}^{x^{3}} 2 \tan t d t $$ $$ \left.A^{\prime}(x)=[?] x\right] \tan x $$

UpStudy AI · Accepted Answer

We start with the function
$$
A(x)=\int_{1}^{x^{3}} 2 \tan t\,dt.
$$
To find the derivative $$ A'(x) $$, we use the Fundamental Theorem of Calculus along with the Chain Rule.

Define
$$
u(x)=x^{3},
$$
so that $$ A(x)=\int_{1}^{u(x)} 2 \tan t\,dt $$.

Taking the derivative, we have
$$
A'(x)=2 \tan(u(x))\cdot u'(x).
$$
Since
$$
u'(x)=\frac{d}{dx}(x^{3})=3x^{2},
$$
we substitute to obtain
$$
A'(x)=2 \tan(x^{3})\cdot(3x^{2}).
$$
Thus,
$$
A'(x)=6x^{2}\tan(x^{3}).
$$
$$ A'(x) = 6x^{2} \tan(x^{3}) $$

Find the derivative and simplify your
answer.

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