Question

Find $d y / d x$. $$y=4 \sqrt{\sec x+\tan x}$$

   Find $d y / d x$.

$$y=4 \sqrt{\sec x+\tan x}$$

Calculus: Graphical, Numerical, Algebraic

Calculus: Graphical, Numerical, Algebraic

Ross L. Finney,… 5th Edition

Chapter 4, Problem 18

Instant Answer

Solved by Expert Adrian Co

08/28/2021

Step 1

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, the outer function is $4\sqrt{x}$ and the inner function is $\sec x + \tan x$. So, we Show more…

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Find $d y / d x$. $$y=4 \sqrt{\sec x+\tan x}$$

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Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. It states that if a function can be expressed as an outer function applied to an inner function, then its derivative is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. This concept is crucial when differentiating expressions where one function is nested inside another.

Derivatives of Trigonometric Functions

Understanding the derivatives of basic trigonometric functions is essential. Specifically, the derivative of the secant function is the product of the secant and tangent functions, and the derivative of the tangent function is the square of the secant function. These standard formulas are applied directly when differentiating expressions that include trigonometric components.

Differentiation of Radical Functions

Radical functions, such as square roots, are differentiated by rewriting them as exponential functions with fractional exponents. The differentiation then follows the power rule, which involves multiplying by the exponent and reducing the exponent by one. This technique is particularly important when the square root is applied to another function, as it must be combined with the chain rule for a complete differentiation.

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