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Given
y=tan^(2)(x), find
(dy)/(dx).
Answer:
(dy)/(dx)=

Given $y=\tan ^{2}(x)$ , find $\frac{d y}{d x}$ . $\newline$ Answer: $\frac{d y}{d x}=$

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Write a formula for an arithmetic sequence

Full solution

Q. Given

y=\tan ^{2}(x)

, find

\frac{d y}{d x}

.

\newline

Answer:

\frac{d y}{d x}=

Identify Function: Identify the function to differentiate. The function given is $y = \tan^2(x)$ , which means $y = (\tan(x))^2$ . We need to find the derivative of this function with respect to $x$ .
Apply Chain Rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is $f(u) = u^2$ and the inner function is $u = \tan(x)$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. The derivative of $f(u) = u^2$ with respect to $u$ is $f'(u) = 2u$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $u = \tan(x)$ with respect to $x$ is $u' = \sec^2(x)$ .
Combine Derivatives: Combine the derivatives using the chain rule. Multiply the derivative of the outer function by the derivative of the inner function to get the derivative of the composite function. This gives us $(\frac{dy}{dx}) = 2 \cdot \tan(x) \cdot \sec^2(x)$ .
Simplify Expression: Simplify the expression if possible. In this case, the expression $2 \cdot \tan(x) \cdot \sec^2(x)$ is already simplified, so we can state the final answer.

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