Differentiation
In addition to solving for derivatives, we can also use many derivative rules to solve for equations. The slope formula, quotient rule and instantaneous rate of change formula can only be used to find derivatives of power and linear functions. There are many rules that we can use to find the derivative of power functions, exponential functions, logarithmic functions, trigonometric functions, product functions, quotient functions, compound functions, and implicit functions. The following slide show gives us a clear layout of the rules and examples of how to use the rules.
The following problem uses multiple derivative rules, including the power, product, exponential, log and trigonometric function rules.
In step one we can simplify the original problem. Since cos(x) over sin(x) equals cot(x), we can multiply the first part of the numerator with cot(x).
Step 2 consists of many smaller steps that lead us to the derivative of function f(x).
We first have to identify our f(x) and g(x) functions. After figuring that out, use derivative rules do differentiate f(x), and multiply your answer with g(x). Next add that entire expression(use parenthesis!) to f(x) times the derivative of g(x). Simplify your answer as much as you can.
Note: The first part of the function(f(x)) requires you to use the chain rule. Therefore you will have to identify the inside and outside function. The inside function is the exponent, and the outside exponent is 5^((x^7)+ex+7ln(x)+π).
Step 2 consists of many smaller steps that lead us to the derivative of function f(x).
We first have to identify our f(x) and g(x) functions. After figuring that out, use derivative rules do differentiate f(x), and multiply your answer with g(x). Next add that entire expression(use parenthesis!) to f(x) times the derivative of g(x). Simplify your answer as much as you can.
Note: The first part of the function(f(x)) requires you to use the chain rule. Therefore you will have to identify the inside and outside function. The inside function is the exponent, and the outside exponent is 5^((x^7)+ex+7ln(x)+π).