9–61. Evaluate and simplify y'.


y = tan (sin θ)

Verified step by step guidance

1. First, identify the function y in terms of θ. Here, y = tan(sin(θ)).
2. To find y', the derivative of y with respect to θ, apply the chain rule. The chain rule states that if you have a composite function f(g(x)), the derivative is f'(g(x)) * g'(x).
3. In this case, let u = sin(θ), so y = tan(u). The derivative of tan(u) with respect to u is sec²(u).
4. Next, find the derivative of u = sin(θ) with respect to θ, which is cos(θ).
5. Combine these results using the chain rule: y' = sec²(sin(θ)) * cos(θ). This is the derivative of y with respect to θ.

Key Concepts

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve of the function at any given point. In this context, we need to apply the rules of differentiation to find the derivative of the function y = tan(sin θ).

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. It states that if a function y is composed of two functions u and v, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. In this case, we will use the chain rule to differentiate tan(sin θ) by first differentiating the outer function (tan) and then the inner function (sin).

Trigonometric Derivatives

Trigonometric derivatives are the derivatives of trigonometric functions, which are essential for solving problems involving angles and periodic functions. For instance, the derivative of tan(x) is sec²(x), and the derivative of sin(x) is cos(x). Understanding these derivatives is crucial for evaluating y' in the given function, as we will need to apply these specific derivatives during the differentiation process.