Question

Find $d y / d x$ $$y=\sqrt[3]{2+\tan \left(x^{2}\right)}$$

   Find $d y / d x$
$$y=\sqrt[3]{2+\tan \left(x^{2}\right)}$$

Calculus A New Horizon

Calculus A New Horizon

Howard Anton 6th Edition

Chapter 4, Problem 2

Instant Answer

Solved by Expert Monica Miller

06/30/2021

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We can rewrite the function as $y=\left(2+\tan \left(x^{2}\right)\right)^{1/3}$. Show more…

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Find $d y / d x$ $$y=\sqrt[3]{2+\tan \left(x^{2}\right)}$$

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Key Concepts

Derivative of Trigonometric Functions

Certain trigonometric functions like tangent have specific derivative formulas; for example, the derivative of tan(u) is sec^2(u) multiplied by the derivative of the argument u. This principle is critical when these functions are part of a composite function.

Chain Rule

The chain rule is a fundamental tool in calculus for differentiating composite functions. It states that if one function is nested inside another, the derivative of the entire expression is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Power Rule for Differentiation

The power rule is used for differentiating functions of the form x^n, including those with fractional exponents such as cube roots. It allows one to compute the derivative by multiplying by the exponent and subtracting one from the exponent in the resulting expression.

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