Questions: Differentiate. f(x)=e^(x^3+10x) f'(x)=

Transcript text: Differentiate. \[ f(x)=e^{x^{3}+10 x} \] \[ f^{\prime}(x)= \]

Solution

Step 1: Define the Function

Let \( f(x) = e^{x^3 + 10x} \).

Step 2: Identify the Inner Function

Define the inner function as \( u = x^3 + 10x \).

Step 3: Differentiate the Inner Function

Calculate the derivative of the inner function: \[ \frac{du}{dx} = 3x^2 + 10 \]

Step 4: Differentiate the Outer Function

Differentiate the outer function with respect to \( u \): \[ \frac{df}{du} = e^u \]

Step 5: Apply the Chain Rule

Using the chain rule, the derivative of \( f(x) \) is: \[ f'(x) = \frac{df}{du} \cdot \frac{du}{dx} = e^{x^3 + 10x} \cdot (3x^2 + 10) \]

Step 6: Write the Final Derivative

Thus, the derivative of the function is: \[ f'(x) = (3x^2 + 10)e^{x^3 + 10x} \]

\(\boxed{f'(x) = (3x^2 + 10)e^{x^3 + 10x}}\)

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