9–61. Evaluate and simplify y'.


y = 10^sin x+sin¹⁰x

Verified step by step guidance

1. Step 1: Identify the function y = 10^{\(\sin\) x} + (\(\sin\) x)^{10}. This function is a sum of two terms: 10^{\(\sin\) x} and (\(\sin\) x)^{10}.
2. Step 2: Differentiate the first term 10^{\(\sin\) x} using the chain rule. The derivative of a^u, where a is a constant and u is a function of x, is a^u \(\ln\)(a) \(\cdot\) u'. Here, a = 10 and u = \(\sin\) x.
3. Step 3: Differentiate \(\sin\) x with respect to x to find u'. The derivative of \(\sin\) x is \(\cos\) x.
4. Step 4: Differentiate the second term (\(\sin\) x)^{10} using the chain rule. The derivative of u^n, where u is a function of x and n is a constant, is n \(\cdot\) u^{n-1} \(\cdot\) u'. Here, u = \(\sin\) x and n = 10.
5. Step 5: Combine the derivatives from Steps 2 and 4 to find y'. The derivative of y is the sum of the derivatives of its terms.

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the given function y = 10^sin x + sin¹⁰x to find y'.

Chain Rule

The Chain Rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, 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. This is particularly relevant for terms like 10^sin x.

Power Rule

The Power Rule is a basic rule for differentiation that states if y = x^n, then the derivative y' = n*x^(n-1). This rule will be applied to the term sin¹⁰x, where we treat sin x as a function raised to a power, allowing us to find its derivative effectively.