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Chapter 3: Problem 57

Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. $$y=e^{x^{2}+1} \sin x^{3}$$

Short Answer

Expert verified
Question: Find the derivative of the function \(y = e^{x^{2}+1} \sin x^{3}\) with respect to x. Answer: The derivative of the given function with respect to x is: \(\frac{dy}{dx} = e^{x^2+1}\left[2x\sin x^3 + 3x^2 \cos x^3\right]\).

Step by step solution

01

Identify the functions

First, we need to identify the main functions involved. The given function is: $$y=e^{x^{2}+1} \sin x^{3}$$ We can see that it is a product of two functions: \(f(x) = e^{x^{2}+1}\) and \(g(x) = \sin x^{3}\) Step 2: Apply the Product Rule
02

Apply the Product Rule

Since the given function is a product of two functions, we can use the Product Rule to differentiate it. The Product Rule states: $$(f(x) \cdot g(x))' = f'(x) g(x) + f(x) g'(x)$$ Now our goal is to find \(f'(x)\) and \(g'(x)\). Step 3: Differentiate f(x)
03

Differentiate f(x)

To find the derivative of \(f(x) = e^{x^{2}+1}\), we need to use the Chain Rule, which states: $$\frac{d}{dx}(u(v(x))) = u'(v(x)) \cdot v'(x)$$ Here, our outer function (\(u(v)\)) is \(e^v\), and our inner function (\(v(x)\)) is \(x^2 + 1\). Differentiating each function gives: $$\frac{d}{dv}(e^v) = e^v$$ $$\frac{d}{dx}(x^2 + 1) = 2x$$ Applying the Chain Rule to find \(f'(x)\), we have: $$f'(x) = e^{x^2+1} \cdot 2x$$ Step 4: Differentiate g(x)
04

Differentiate g(x)

similar to step 3, find the derivative of \(g(x) = \sin x^{3}\). For this, let's use the Chain Rule. Outer function (\(u(v)\)) is \(\sin v\) and the inner function (\(v(x)\)) is \(x^3\). Differentiating each function gives: $$\frac{d}{dv}(\sin v)=\cos v$$ $$\frac{d}{dx}(x^3) = 3x^2$$ Applying the Chain Rule to find \(g'(x)\), we have: $$g'(x) = \cos x^3 \cdot 3x^2$$ Step 5: Combine the results
05

Combine the results

Now, let's substitute \(f'(x)\) and \(g'(x)\) back into the Product Rule formula to find the derivative of the given function: $$(e^{x^2+1} \sin x^3)' = f'(x) g(x) + f(x) g'(x)$$ Substitute the values: $$ = (e^{x^2+1}2x) \sin x^3 + e^{x^2+1} (\cos x^3\cdot3x^2)$$ Step 6: Simplify the expression
06

Simplify the expression

Now let's simplify the expression by factoring out common terms: $$ = e^{x^2+1}\left[2x\sin x^3 + 3x^2 \cos x^3\right]$$ The final expression for the derivative is: $$\frac{dy}{dx} = e^{x^2+1}\left[2x\sin x^3 + 3x^2 \cos x^3\right]$$

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