How to Differentiate With Respect to X

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Taking the derivative of a function tells you something about how it behaves. A function is made up out of numbers and variables, added and factored together in different combinations. When you calculate the derivative of a function, or differentiate it, you see how that function changes when its variables change. To differentiate a function with respect to the variable x, for example, you must simply make a few changes to each term.

  • Erase the first term if it does not have an x in it. Terms are numbers, variables and expressions multiplied together. They are separated from each other by addition and subtraction signs. For example, if your first term is 43y, then erase it. Erase every other term with no x in it.

  • Take the derivative of the first term that is a function of x. To do this, multiply the first term of the function by the exponent of the x, then subtract one from that exponent. For example, if the term is abx^3, then you multiply the whole term by three, which is the exponent of x, then subtract one from three, resulting in 3abx^2. Adjust every other term in the same manner. These first two steps are the most basic form of taking the derivative of a term with respect to x.

  • Take the derivative of the first term that is made of two functions of x multiplied together. To do this, first take the derivatives of each of the two functions and record them. Next, multiply the first function by the derivative of the second function. Multiply the second function by the derivative of the first. Add the products together. Adjust all other terms made of two functions of x multiplied together in the same manner.

  • Take the derivative of the first term that is made of one function of x divided by another. Take the derivative of both the numerator and the denominator. Multiply the denominator by the derivative of the numerator. Call this "product one." Multiply the numerator by the derivative of the denominator. Call this "product two." Subtract product two from product one. Divide this result by the square of the denominator.

  • Take the derivative of trigonometric functions by replacing each sin(x) with a cos(x), each cos(x) with a --sin(x), and each tan(x) with a sec^2(x). Replace each cot(x) with a --csc^2(x), each sec(x) with a sec(x)tan(x) and every csc(x) with a --csc(x)cot(x).

  • Replace every natural logarithm with the expression 1/x. Leave any constant e raised to the x power alone. Though it may seem counterintuitive, the derivative of e^x is e^x. Multiply any other number or variable raised to the x power by the natural logarithm of that number.

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