Find the derivative of f(x)=6sec(x)+x^(3).

Identify Functions: Identify the individual functions within f(x) that we need to differentiate. f(x) = 6sec(x) + x^3 consists of two terms: 6sec(x) and x^3. We will differentiate each term separately.Differentiate First Term: Differentiate the first term, 6sec(x). The derivative of sec(x) is sec(x)tan(x), so the derivative of 6sec(x) is 6 * sec(x)tan(x).Differentiate Second Term: Differentiate the second term, x^3. The derivative of x^3, using the power rule, is 3x^2.Combine Derivatives: Combine the derivatives of the individual terms to find the derivative of f(x). f'(x) = derivative of 6sec(x) + derivative of x^3 f'(x) = 6 * sec(x)tan(x) + 3x^2Check for Errors: Check for any mathematical errors in the differentiation process. No errors were made in applying the differentiation rules for sec(x) and x^3.

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Q. Find the derivative of

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f(x)=6\sec(x)+x^{3}.

Identify Functions: Identify the individual functions within $f(x)$ that we need to differentiate. $f(x) = 6\sec(x) + x^3$ consists of two terms: $6\sec(x)$ and $x^3$ . We will differentiate each term separately.
Differentiate First Term: Differentiate the first term, $6\sec(x)$ . The derivative of $\sec(x)$ is $\sec(x)\tan(x)$ , so the derivative of $6\sec(x)$ is $6 \times \sec(x)\tan(x)$ .
Differentiate Second Term: Differentiate the second term, $x^3$ . The derivative of $x^3$ , using the power rule, is $3x^2$ .
Combine Derivatives: Combine the derivatives of the individual terms to find the derivative of $f(x)$ . $f'(x) = \text{derivative of } 6\sec(x) + \text{derivative of } x^3$ $f'(x) = 6 \cdot \sec(x)\tan(x) + 3x^2$
Check for Errors: Check for any mathematical errors in the differentiation process. $\newline$ No errors were made in applying the differentiation rules for $\sec(x)$ and $x^3$ .