Let `f(x)=3x^(3)+7x^(5)`. Find `f^(')(2)`.

Answer

To find the derivative of the function \( f(x) = 3x^3 + 7x^5 \) and evaluate it at \( x = 2 \), we will follow these steps:

### Step 1: Differentiate the function
To find \( f'(x) \), we will use the power rule of differentiation. The power rule states that if \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{n-1} \).

Applying this to each term in \( f(x) \):

1. For the first term \( 3x^3 \):
\[ \frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2 \]

2. For the second term \( 7x^5 \):
\[ \frac{d}{dx}(7x^5) = 5 \cdot 7x^{5-1} = 35x^4 \]

Combining these results, we get:
\[ f'(x) = 9x^2 + 35x^4 \]

### Step 2: Evaluate the derivative at \( x = 2 \)
Now we need to find \( f'(2) \):
\[ f'(2) = 9(2^2) + 35(2^4) \]

Calculating each term:
1. Calculate \( 2^2 \):
\[ 2^2 = 4 \quad \Rightarrow \quad 9(2^2) = 9 \cdot 4 = 36 \]

2. Calculate \( 2^4 \):
\[ 2^4 = 16 \quad \Rightarrow \quad 35(2^4) = 35 \cdot 16 = 560 \]

Now, combine these results:
\[ f'(2) = 36 + 560 = 596 \]

### Final Answer
Thus, the value of \( f'(2) \) is \( 596 \).

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