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Chapter 3: Problem 53

Calculate the derivative of the following functions. $$y=\sin \left(\sin \left(e^{x}\right)\right)$$

Short Answer

Expert verified
Question: Find the derivative of the function \(y = \sin(\sin(e^{x}))\) with respect to x. Answer: The derivative of the given function is $$y' = \cos\left(\sin\left(e^{x}\right)\right)\cdot \cos\left(e^{x}\right)\cdot e^{x}$$

Step by step solution

01

Identify the function composition

The given function is a composition of three functions: sine function, sine function again, and the exponential function. Let's identify these functions: 1. \(f(x) = \sin(x)\) 2. \(g(x) = \sin(x)\) 3. \(h(x) = e^{x}\) Our composite function \(y\) can be expressed as \(y = f(g(h(x)))\).
02

Apply the chain rule

We need to apply the chain rule to find the derivative of the composite function \(y\). The chain rule states that if we have a composite function as \(y = f(g(h(x)))\), then its derivative can be found using the following formula: \(y' = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)\)
03

Find the derivatives of individual functions

Now, we need to find the derivatives of the individual functions \(f(x)\), \(g(x)\), and \(h(x)\) that we identified in Step 1. 1. \(f'(x) = \frac{d}{dx}(\sin(x)) = \cos(x)\) 2. \(g'(x) = \frac{d}{dx}(\sin(x)) = \cos(x)\) 3. \(h'(x) = \frac{d}{dx}(e^{x}) = e^{x}\)
04

Substitute the derivatives in the chain rule formula

Now let's substitute the derivatives we found in Step 3 into the chain rule formula: \(y' = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)\) \(y' = \cos(g(h(x))) \cdot \cos(h(x)) \cdot e^{x}\)
05

Replace g(h(x)) and h(x) with the original functions

Now we need to replace \(g(h(x))\) and \(h(x)\) with their original functions to get the final derivative of \(y\): \(y' = \cos(\sin(e^{x}))\cdot\cos(e^{x})\cdot e^{x}\)
06

Final Solution

The derivative of the given function is $$y' = \cos\left(\sin\left(e^{x}\right)\right)\cdot \cos\left(e^{x}\right)\cdot e^{x}$$

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