Simple Differentiation

In calculus the derivative is defined as a measure of a function that changes as its input changes. A simple derivative can be stated as change in one quantity that response to changes in some other quantity. The derivative of a simple function at a certain input describes the preeminent linear approximation of the function close to that input value. In advanced calculus, the derivative of a function at a point is a linear transformation called the linearization. Simple differentiation problem includes one or two step differentiation x, y terms.

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Differentiation and the derivative

A method to compute the immediate rate of change of a function with respect to one of its variables is known as Differentiation. This rate of change is called the derivative of y with value to x. In more exact language, the dependence of y upon x means that y is a function of x. This functional relationship is frequently denoted y = ƒ(x), where ƒ denotes the function. If x and y are real numbers, and if the graph of y is plotted beside x, the derivative measures the slope of this graph at each point.

Linear function is defined as the x – axis linearly varies with y – axis. In this case, y = ƒ(x) = m x + c, for real numbers m and c, and the slope m.

The differentiation is denoted by the symbol as shown below,

x = dx

y = dy

Problems:

Simple differentiation Example 1:

Differentiate f(x) = x2

Solution:

Here f(x) = y

Y= x2

On differentiating this,

dy/dx = 2x


Simple differentiation Example 2:

Differentiate y = 2x3 + 6x2 + 2x

Solution:

Differentiating the above expression with respect to x

dy/dx = 3(2)x2 + 2(6)x + 2

dy/dx = 6x2 + 12x + 2


Simple differentiation Example 3:

Differentiate f (x) = ln x2 +5x

Solution:

f (x) = ln (x2 +5x)

f’(x) = 2x + 5

x2 +5x

The answer is dy/dx = 2x + 5

x2 + 5x

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Simple differentiation Example 4:

Differentiate y = log x + x2 + 2

Solution:

On differentiating log x, we get

Log x = 1/x.

dy/dx = (1/x) + 2x