2.7 Derivatives of cos x, sinx, e^x, and ln x

The derivatives of the trigonometric functions cos x, sin x, and the exponential function e^x, and the natural logarithm function ln x are important concepts in Calculus. These functions have specific rules for finding their derivatives, which make it easier to find the derivative of more complex functions that involve them. The derivative of cos x is -sin x. This means that if we have a function f(x) = cos x, the derivative of the function is f'(x) = -sin x. For example, the derivative of f(x) = cos x is -sin x. The derivative of sin x is cos x. This means that if we have a function f(x) = sin x, the derivative of the function is f'(x) = cos x. For example, the derivative of f(x) = sin x is cos x. The derivative of e^x is e^x. This means that if we have a function f(x) = e^x, the derivative of the function is f'(x) = e^x. For example, the derivative of f(x) = e^x is e^x. The derivative of ln x is 1/x. This means that if we have a function f(x) = ln x, the derivative of the function is f'(x) = 1/x. For example, the derivative of f(x) = ln x is 1/x. It is important to note that these rules are only applicable when the exponent of x is a constant. If the exponent is a variable or a function, different methods such as the chain rule must be used to find the derivative.

Example 1: Find the derivative of f(x) = cos 2x Solution: Using the chain rule, we know that the derivative of a function composed of another function is equal to the derivative of the outer function times the derivative of the inner function. In this case, the function can be written as f(x) = cos (2x) = cos (2x) . The derivative of the inner function 2x is 2. The derivative of the outer function cos(2x) is -sin (2x) so the derivative of f(x) = -sin (2x) * 2 = -2sin (2x) Example 2: Find the derivative of f(x) = sin 3x Solution: Using the chain rule, we know that the derivative of a function composed of another function is equal to the derivative of the outer function times the derivative of the inner function. In this case, the function can be written as f(x) = sin (3x) . The derivative of the inner function 3x is 3. The derivative of the outer function sin(3x) is cos (3x) so the derivative of f(x) = cos (3x) * 3 = 3cos (3x) Example 3: Find the derivative of f(x) = e^(2x) Solution: Using the chain rule, we know that the derivative of a function composed of another function is equal to the derivative of the outer function times the derivative of the inner function. In this case, the function can be written as f(x) = e^(2x) . The derivative of the inner function 2x is 2. The derivative of the outer function e^(2x) is e^(2x) so the derivative of f(x) = e^(2x) * 2 = 2e^(2x)

Example 4:

Find the derivative of f(x) = ln(3x)

Solution: Using the chain rule, we know that the derivative of a function composed of another function is equal to the derivative of the outer function times the derivative of the inner function. In this case, the function can be written as f(x) = ln(3x) . The derivative of the inner function 3x is 3. The derivative of the outer function ln(3x) is 1/3x so the derivative of f(x) = 3/3x = 1/x

Example 5:

Find the derivative of f(x) = e^(sin x)

Solution: Using the chain rule, we know that the derivative of a function composed of another function is equal to the derivative of the outer function times the derivative of the inner function. In this case, the function can be written as f(x) = e^(sin x) . The derivative of the inner function sin x is cos x. The derivative of the outer function e^(sin x) is e^(sin x) so the derivative of f(x) = e^(sin x) * cos x = e^(sin x)cos x

In conclusion, the derivatives of the trigonometric functions cos x, sin x, and the exponential function e^x, and the natural logarithm function ln x are fundamental concepts in Calculus. These functions have specific rules for finding their derivatives, which make it easier to find the derivative of more complex functions that involve them. However, it is important to remember that these rules are only applicable when the exponent of x is a constant. If the exponent is a variable or a function, different methods such as the chain rule must be used to find the derivative.