5. Find the equation of the tangent line to the graph of f(a) = -sec(a) at the point (0, -1)

6. Differentiate y = sin(x)tan(x).

7. Find f '(x) for f(x) = sin(x)cot(x).

8. Find the derivative of the function f(x) = cos^2(x) + tan^2(x)

9. Find f '(x) for f(x) = xln(x) - 5ex.

To find the equation of the tangent line to the graph of f(a) = -sec(a) at the point (0, -1), you can follow these steps:

Step 1: Start by finding the derivative of the function f(a) = -sec(a). The derivative of sec(a) is -sec(a)tan(a).

Step 2: Evaluate the derivative at the given point (0, -1) by substituting a = 0 into the derivative. The value of the derivative at a = 0 is -sec(0)tan(0) = -1(0) = 0.

Step 3: Use the point-slope form of the equation of a line, y - y1 = m(x - x1), where (x1, y1) represents the given point (0, -1) and m represents the slope of the tangent line. Since the slope is 0, the equation becomes y - (-1) = 0(x - 0), which simplifies to y + 1 = 0.

Therefore, the equation of the tangent line to the graph of f(a) = -sec(a) at the point (0, -1) is y = -1.

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To differentiate y = sin(x)tan(x), you can follow these steps:

Step 1: Apply the product rule of differentiation, which states that the derivative of the product of two functions u(x) and v(x) is given by the formula (u'v + uv').

Step 2: Let u(x) = sin(x) and v(x) = tan(x).

Step 3: Find the derivatives of u(x) and v(x). The derivative of sin(x) is cos(x), and the derivative of tan(x) is sec^2(x).

Step 4: Apply the product rule to differentiate y = sin(x)tan(x). The derivative is (cos(x)tan(x) + sin(x)sec^2(x)).

Therefore, the derivative of y = sin(x)tan(x) is cos(x)tan(x) + sin(x)sec^2(x).

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To find f'(x) for f(x) = sin(x)cot(x), you can follow these steps:

Step 1: Start by rewriting cot(x) as cos(x)/sin(x) to simplify the expression.

Step 2: Multiply sin(x) and cos(x)/sin(x) together to get cos(x).

Step 3: The derivative of cos(x) is -sin(x).

Therefore, the derivative of f(x) = sin(x)cot(x) is -sin(x).

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To find the derivative of the function f(x) = cos^2(x) + tan^2(x), you can follow these steps:

Step 1: Apply the power rule of differentiation, which states that the derivative of x^n is n*x^(n-1), where n represents any real number.

Step 2: Differentiate cos^2(x) using the chain rule. The derivative of cos^2(x) is 2cos(x)(-sin(x)) = -2sin(x)cos(x).

Step 3: Differentiate tan^2(x) using the chain rule. The derivative of tan^2(x) is 2tan(x)(sec^2(x)) = 2sec^2(x)tan(x).

Step 4: Add the derivatives of cos^2(x) and tan^2(x) together. The derivative of f(x) = cos^2(x) + tan^2(x) is -2sin(x)cos(x) + 2sec^2(x)tan(x).

Therefore, the derivative of f(x) = cos^2(x) + tan^2(x) is -2sin(x)cos(x) + 2sec^2(x)tan(x).

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To find f'(x) for f(x) = xln(x) - 5ex, you can follow these steps:

Step 1: Apply the product rule of differentiation, which states that the derivative of the product of two functions u(x) and v(x) is given by the formula (u'v + uv').

Step 2: Let u(x) = x and v(x) = ln(x) - 5ex.

Step 3: Find the derivatives of u(x) and v(x). The derivative of x is 1, and the derivative of ln(x) - 5ex can be found by differentiating each term separately. The derivative of ln(x) is 1/x, and the derivative of -5ex is -5ex.

Step 4: Apply the product rule to differentiate f(x) = xln(x) - 5ex. The derivative is (1)(ln(x) - 5ex) + (x)(1/x - 5ex).

Step 5: Simplify the expression. The derivative of f(x) = xln(x) - 5ex is ln(x) - 5ex + 1 - 5ex.

Therefore, the derivative of f(x) = xln(x) - 5ex is ln(x) - 10ex + 1.