A derivative is a branch of mathematics, which is about measuring how to function the changes and the input changes. One can think of a derivative as how much one quantity keeps changing in response to other changing quantities. The derivate at a certain function will describe the best approximation for linear, which is near the input value. Taking derivatives of integrals is very easy if you keep some basic rules in mind. Here are some simple steps on how to take derivatives.
Things Required:
- Calculator
Simple Exponential Expressions
- First, you will have to identify the exponent on the variable for which you need to take the derivative. If you read the problem carefully, then you will be able to figure out, which variable you must use. You can also look at the bottom of the d/dx fraction. Whatever is placed in the place of ‘x’ inside the fraction that is the variable you need to look for. If there is no exponent written, then the value of the exponent will be one. If the variable is present at the bottom of the fraction, then you will have to move the variable on the top and take the exponent as negative.
- Now, you will have to multiply the term by the value of the exponent. Remember to reduce the value of exponent by 1. Therefore, if the exponent of the integral was given as three, the exponent for the derivative should be two. If the exponent of the integral was given as -3, then the exponent of the derivative will be -4. If the exponent in the integral was given as one, then the derivative will be 0, which will make the variable value one. You will have to simplify the term. If there is more than one term, then you must always take the derivative by itself.
Natural logarithms and Trigonometric functions
- First, you will have to check which of the trigonometric functions you have to find the derivative of. You will have to apply the following rules for the derivatives:
- the derivative for sin(x) will be cos(x);
- for cos(x) it is –sin(x);
- for tan(x) it is sec^2(x);
- for cot(x) it is -csc^2(x);
- for sec(x) it is tan(x);
- for csc(x) it is –csc(x)cot(x).
Always remember that the derivative for all the ‘co-‘ will be negative.
- You can apply this simple rule to the natural log, which is In(x) is always 1/x. then you will have to simplify the expression.
Terms where the variable can occur more than once
- Start with breaking the compound functions to the individual parts. Find every individual place where the integral has occurred. Now, you will have to take the derivative to the first function and then you can add it to other functions. After this take the derivative to the second function and add to the other functions.
- You will have to continue with this process until you have derived all the functions. In the end you will get various terms in the derivatives as you had different functions in the same term of the integral.
Compound terms: Chain Rule
- Start with identifying each function inside the term of the integral. Now, you will have to take the derivative of the outside function and keep the original function the way it was. After this, you will have to multiply the derivative of the outside function with the derivative present on the inside function. Then you can simplify the expression.
Tips and Warnings:
- Remember that you will always have to keep a track of the negative signs. You must write every problem before you start taking the derivative. Write the entire problem and this will force you to notice everything that you need to know about the problem.
- You should know that the derivative of yz (where z and y are both separate functions) is not 1, as y and z are two different functions. For this, you will have to use the product rule, yz = y(1) + z(1) = y + z.
- You will have to keep practicing the quotient rule, product rule and the implicit differentiation as these are the most difficult ones in calculus. Try to memorize the basic trig derivatives and learn how to manipulate them.
- Always remember that the minus sign in front of ‘f’ whenever you are using the quotient rule, this is one of the most common mistakes that everyone makes.
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