In Exercises 41–58, find dy/dt.


y = (t tan(t))¹⁰

Verified step by step guidance

1. Step 1: Recognize that you need to find the derivative of y with respect to t, where y = (t tan(t))^10. This involves using the chain rule and the product rule.
2. Step 2: Apply the chain rule. Let u = t tan(t), so y = u^10. The derivative dy/dt = 10u^9 * du/dt.
3. Step 3: Find du/dt using the product rule. Since u = t tan(t), du/dt = d(t tan(t))/dt = t * d(tan(t))/dt + tan(t) * d(t)/dt.
4. Step 4: Calculate the derivatives: d(tan(t))/dt = sec^2(t) and d(t)/dt = 1. Substitute these into the expression for du/dt to get du/dt = t * sec^2(t) + tan(t).
5. Step 5: Substitute du/dt back into the expression for dy/dt: dy/dt = 10 * (t tan(t))^9 * (t sec^2(t) + tan(t)).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(t)) is composed of two functions, the derivative dy/dt is found by multiplying the derivative of the outer function f with respect to the inner function g by the derivative of the inner function g with respect to t. This rule is essential for differentiating expressions like y = (t tan(t))¹⁰.

Product Rule

The product rule is used to differentiate functions that are the product of two or more functions. If y = u(t) * v(t), then the derivative dy/dt is u'(t)v(t) + u(t)v'(t). In the given problem, t and tan(t) are multiplied, so the product rule helps in finding the derivative of the inner function t tan(t) before applying the chain rule.

Trigonometric Derivatives

Understanding the derivatives of trigonometric functions is crucial for solving calculus problems involving trigonometric expressions. The derivative of tan(t) is sec²(t), which is necessary when applying the product rule to differentiate t tan(t). This knowledge allows for the correct application of differentiation rules to trigonometric components within a function.