Studying a function is to study the properties and image characteristics of this function. The important properties of a function mainly involve monotonicity, parity, symmetry, maximum, extreme value, extreme point, zero, domain, range, tangent, asymptote and so on.

Derivative, as an important tool to study functions, is mainly used to help study monotonicity, extreme value, maximum value and tangent of functions, find asymptotes perpendicular to the X axis, and draw general images of functions by combining the positive and negative values of derivatives. Let's briefly introduce them respectively.

Firstly, the monotonicity of differentiable functions is studied.

If a function is differentiable, according to the positive and negative values of the derivative, there are roughly the following situations in a certain interval.

1. The derivative function value is always greater than or equal to zero (the number of points with the derivative value equal to zero is limited). At this point, the original function is a strict increasing function in this interval.

2. The derivative function value is always less than or equal to zero (the number of points with derivative value equal to zero is limited). At this point, the original function is a strictly decreasing function in this interval.

3. The value of derivative function is always zero. At this point, the original function is a constant function in this interval.

Secondly, the extreme value of differentiable function is studied.

The extreme value of a function is different from the maximum value of the function, and the maximum value describes the properties of the function on the local part (except the two ends and the discontinuous points).

1. If the left derivative value of the function at local point P is greater than 0 and the right derivative value is less than 0, the original function "increases left and decreases right" near point P, and the function image is the highest near point P, and the function value corresponding to point P is the maximum value of the function. Meanwhile, the abscissa of point P is called the maximum point of the function.

2. If the left derivative value of the function at a local point P is less than 0 and the right derivative value is greater than 0, and the original function "decreases left and increases right" near point P, then the function image is the highest near point P, and the function value corresponding to point P is the maximum value of the function. Meanwhile, the abscissa of point P is called the maximum point of the function.

Thirdly, the maximum and range of differentiable functions are studied.

Finding the range of function is inseparable from finding the maximum value of function, and the maximum value is closely related to the extreme value of derivative function. The differences and connections between the extreme value of a function and the maximum value of a function are mainly as follows.

1. If the function is differentiable, the extreme value of the function is not necessarily the extreme value of the function, and the extreme value of the function is not necessarily the extreme value of the function.

2. The maximum value of a differentiable function is often obtained from all the extreme values of this function and the functions at the end points.

3. If the maximum value of a differentiable function is obtained within its definition domain (not at the endpoints and discontinuous points), then the maximum value of the function at this time must also be an extreme value of the function.

Based on the above facts, we only need to find the function value at the end of the interval, the function value of the discontinuous point and all the extreme values of the differentiable function when we find the maximum value of a differentiable function, and then compare them. Where the maximum value is the maximum value of the function and the minimum value is the minimum value of the function.

Fourthly, the tangent of differentiable function is studied.

To solve the tangent equation of differentiable function, the slope of this tangent is often required. The relationship between the derivative value of the derivative function and the tangent slope is that the derivative value of the derivative function at the tangent point is equal to the slope value of the tangent at the tangent point. Therefore, when calculating the slope of the tangent, we only need to find the derivative function of the original function first, and then find the derivative value corresponding to the derivative function at the tangent point.

Fifthly, the asymptotes of differentiable functions are studied.

Using the derivative tool to directly find the asymptote of a function is mainly to find the asymptote perpendicular to the X axis in the coordinate system. Because the slope of this asymptote does not exist, the corresponding derivative function value is taken as "infinity" (positive infinity or negative infinity) here.

Based on the above principle, when finding the asymptote of the function image without knowing it, after finding its derivative function, the point where the derivative function is infinite can be found.

Sixth, combine the positive and negative derivative values to draw an overview of the function.

After finding the definition domain of the function, using the derivative of the function as a tool, the extreme point, zero point and endpoint of the function are found, and the monotone interval of the function is judged by combining the positive and negative derivatives, and then the rough image of the function is obtained by connecting these points with smooth curves.

The function of drawing with derivative is very powerful, which can help us draw many function images other than common basic functions.