Questions: D(x)=6(x^2+e^x)^12

Transcript text: $D(x)=6\left(x^{2}+e^{x}\right)^{12}$

Solution

Solution Steps

To find the derivative of the function $ D(x) = 6(x^2 + e^x)^{12} $, we will use the chain rule and the power rule. The chain rule is used to differentiate composite functions, and the power rule is used to differentiate functions of the form $ u^n $. First, differentiate the outer function, then multiply by the derivative of the inner function.

Step 1: Define the Function

We start with the function defined as: \[ D(x) = 6\left(x^2 + e^x\right)^{12} \]

Step 2: Apply the Chain Rule

To find the derivative $ D'(x) $, we apply the chain rule. The derivative of $ u^n $ is $ n \cdot u^{n-1} \cdot u' $, where $ u = x^2 + e^x $ and $ n = 12 $.

Step 3: Differentiate the Inner Function

The inner function $ u = x^2 + e^x $ has a derivative: \[ u' = \frac{d}{dx}(x^2) + \frac{d}{dx}(e^x) = 2x + e^x \]

Step 4: Combine the Results

Using the chain rule, we find: \[ D'(x) = 6 \cdot 12 \cdot (x^2 + e^x)^{11} \cdot (2x + e^x) \] This simplifies to: \[ D'(x) = 72(2x + e^x)(x^2 + e^x)^{11} \]

Final Answer

The derivative of the function is: \[ \boxed{D'(x) = 72(2x + e^x)(x^2 + e^x)^{11}} \]

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