Post 5: Trigonometric and Exponential Functions

Trigonometric functions and exponential functions are commonly encountered in calculus, and it is crucial to understand their derivative rules for solving problems involving these functions. In this post, we will discuss the derivative rules for trigonometric functions like sine, cosine, and tangent, as well as exponential functions, including the natural exponential function e^x.

1. Derivative Rules for Trigonometric Functions:

a) Derivative of Sine (sin(x)): The derivative of sine is cosine. This can be written as: d/dx(sin(x)) = cos(x)

b) Derivative of Cosine (cos(x)): The derivative of cosine is negative sine. This can be written as: d/dx(cos(x)) = -sin(x)

c) Derivative of Tangent (tan(x)): The derivative of tangent can be found using the quotient rule. It can be written as: d/dx(tan(x)) = sec^2(x), where sec(x) = 1/cos(x)

2. Derivative Rules for Exponential Functions:

a) Derivative of the Natural Exponential Function (e^x): The derivative of e^x is simply e^x. This can be written as: d/dx(e^x) = e^x

b) Derivative of Exponential Functions with a Base Other than e: If we have an exponential function of the form a^x, where 'a' is a constant greater than 0 and not equal to 1, the derivative can be found using the chain rule. It can be written as: d/dx(a^x) = ln(a) * a^x

Example 1: Find the derivative of the function f(x) = sin(x) + e^x.

Solution: To find the derivative of f(x), we will apply the derivative rules discussed earlier.

d/dx(sin(x)) = cos(x) d/dx(e^x) = e^x

Using the sum rule for derivatives, the derivative of f(x) can be found as:

f'(x) = d/dx(sin(x)) + d/dx(e^x) = cos(x) + e^x

Example 2: Find the derivative of the function g(x) = 3cos(x) - 2e^x.

Solution: To find the derivative of g(x), we will apply the derivative rules discussed earlier.

d/dx(cos(x)) = -sin(x) d/dx(e^x) = e^x

Using the sum rule for derivatives, the derivative of g(x) can be found as:

g'(x) = 3 * d/dx(cos(x)) - 2 * d/dx(e^x) = 3 * (-sin(x)) - 2 * e^x = -3sin(x) - 2e^x

By understanding these derivative rules for trigonometric and exponential functions, you will be able to find the derivatives of more complex functions involving these functions. Remember to practice these rules and apply them to various examples to strengthen your understanding.