A derivative is one of the major branches of calculus that is essential for determining the change rate of the function according to the independent variable. Calculus is a mathematical branch that allows us to deal with the changes and motions of functions.
The derivative and integral are the main kinds of calculus that deal with the rate of change and finding the area under the curves respectively. In this post, we will explore the basics of the term derivative calculus along with its calculations.
Derivative Calculus
A derivative is the sub-type of calculus that is used to find the slope of the curves and the rate of change of functions with respect to the independent variable. Algebraically, this branch of calculus is frequently used to find the instantaneous rate of change functions.
Geometrically, this branch of calculus is used to find the slope of the tangent line to make the graph of that function at a particular point. Generally, this branch is used to know the changes that occur in the output as the input changes slightly.
If the function “h(v)” is given, then the differential of that function with respect to “v” is represented as h’(v) or dy/dv.
What is the Instantaneous Rate of Change?
In derivative calculus, the concept of rate of change is frequently used to understand how the given function “h(v)” behaves at a particular point on its curve. The rate at which the output of the function is changing at the exact point is also determined with the help of the instantaneous rate of change.
Such as the Instantaneous Rate of Change of a function “h(v)” at a specific point v = b is given by the differential of that function h’(b) that is calculated at v = b.
Manual Evaluation of Derivatives
Below are a few ways to evaluate the derivatives manually.
The Limit Definition of a Derivative
The limit concept could be used to define the derivative of the function p(v) at the particular point v = b. The differential represents the instantaneous rate of change of the function at that particular point “b”. The formula for the limit definition of a derivative is:
p’(v) = limh→0 [(p(v + h) – p(v)) / h]
Where “h” is the minor increment in “v”.
Follow the below steps to evaluate the derivatives according to a limit method.
- First of all, replace p(v) with the given function.
- Put (v + h) in the place of “v” in that function.
- Take a difference of p(v + h) and p(v).
- Divide the subtracted result by “h”.
- Take the limit of the final expression.
Here is a solved example to understand this concept accurately.
Example
Find the derivative of v2 – 4v + 12 with respect to “v” according to the limit definition of the derivative.
Solution
Step 1: First of all, replace p(v) with the given function.
p(v) = v2 – 4v + 12
Step 2: Put (v + h) in the place of “v” in that function.
p(v + h) = (v + h)2 – 4(v + h) + 12
p(v + h) = v2 + 2hv + h2 – 4v – 4h + 12
Step 3: Take a difference of p(v + h) and p(v).
p(v + h) – p(v) = v2 + 2hv + h2 – 4v – 4h + 12 – (v2 – 4v + 12)
p(v + h) – p(v) = v2 + 2hv + h2 – 4v – 4h + 12 – v2 + 4v – 12
p(v + h) – p(v) = 2hv + h2 – 4h
Step 4: Divide the subtracted result by “h”.
(p(v + h) – p(v)) / h = (2hv + h2 – 4h) / h
(p(v + h) – p(v)) / h = 2hv/h + h2/h – 4h/ h
(p(v + h) – p(v)) / h = 2v + h – 4
Step 5: Now take the general expression of the limit definition of derivative, place the values, and take the limit of the function.
p’(v) = limh→0 [(p(v + h) – p(v)) / h]
d/dv [v2 – 4v + 12] = limh→0 [2v + h – 4]
d/dv [v2 – 4v + 12] = limh→0 [2v] + limh→0 [h] – limh→0 [4]
d/dv [v2 – 4v + 12] = 2v + 0 – 4
d/dv [v2 – 4v + 12] = 2v – 4
Evaluation of derivative with its rules & formulas
Differential rules:
- Sum Rule: d/dv [f(v) + h(v)] = d/dv [f(v)] + d/dv [h(v)]
- Difference Rule: d/dv [f(v) – h(v)] = d/dv [f(v)] – d/dv [h(v)]
- Product Rule: d/dv [f(v) * h(v)] = h(v) d/dv [f(v)] + f(v) d/dv [h(v)]
- Constant Rule: d/dv [K] = 0
- Constant function Rule: d/dv [K f(v)] = K d/dv [f(v)]
- Power Rule: d/dv [f(v)]n = n [f(v)]n-1 d/dv [f(v)]
- Quotient Rule: d/dv [f(v) / h(v)] = 1/[h(v)]2 [h(v) d/dv (f(v)) – f(v) d/dv (h(v))]
Differential formulas:
- d/dv (sin(v)) = cos(v)
- d/dv (cos(v)) = -sin(v)
- d/dv (e^v) = e^v
The above rules and formulas plays an essential role to find the derivative of the given function.
Example
Evaluate the differential of “p(v) = 6v3 – 2v4 + 5cos(v) – (12v2 x v3) with respect to “v”.
Solution
Step 1: First of all, rewrite the given function and apply the notation of differentiation to it.
p(v) = 6v3 – 2v4 + 5cos(v) – (12v2 x v3)
p’(v) = d/dv [6v3 – 2v4 + 5cos(v) – (12v2 x v3)]
Step 2: Now use the law of addition and subtraction of differential calculus.
d/dv [6v3 – 2v4 + 5cos(v) – (12v2 x v3)] = d/dv [6v3] – d/dv [2v4] + d/dv [5cos(v)] – d/dv [(12v2 x v3)]
Step 3: Now apply the product law to the above expression
d/dv [6v3 – 2v4 + 5cos(v) – (12v2 x v3)] = d/dv [6v3] – d/dv [2v4] + d/dv [5cos(v)] – [d/dv (12v2) x v3 + 12v2 x d/dv [v3]]
d/dv [6v3 – 2v4 + 5cos(v) – (12v2 x v3)] = d/dv [6v3] – d/dv [2v4] + d/dv [5cos(v)] – d/dv (12v2) x v3 – 12v2 x d/dv [v3]
Step 4: Now take the constant coefficient outside the notation.
d/dv [6v3 – 2v4 + 5cos(v) – (12v2 x v3)] = 6d/dv [v3] – 2d/dv [v4] + 5d/dv [cos(v)] – 12v3 d/dv [v2] – 12v2 d/dv [v3]
Step 5: Now use the power rule to differentiate the above expression.
d/dv [6v3 – 2v4 + 5cos(v) – (12v2 x v3)] = 6 [3 v3 – 1] – 2 [5 v4 – 1] + 5 [-sin(v)] – 12v3 [2 v2 – 1] – 12v2 [3 v3 – 1]
d/dv [6v3 – 2v4 + 5cos(v) – (12v2 x v3)] = 6 [3 v2] – 2 [5 v3] + 5 [-sin(v)] – 12v3 [2 v1] – 12v2 [3 v2]
d/dv [6v3 – 2v4 + 5cos(v) – (12v2 x v3)] = 6 [3 v2] – 2 [5 v3] – 5 [sin(v)] – 12v3 [2 v] – 12v2 [3 v2]
d/dv [6v3 – 2v4 + 5cos(v) – (12v2 x v3)] = 18 v2 – 10 v3 – 5 sin(v) – 24v4 – 36v4
d/dv [6v3 – 2v4 + 5cos(v) – (12v2 x v3)] = 18v2 – 10v3 – 5 sin(v) – 60v4
Evaluation of derivative using Calculator
There are several online calculators that are available to evaluate the derivative of a function with a step by step solution. Here we use a derivative calculator by AllMath and found the results with steps according to different methods.
Let us solve the above example using a calculator.
Step 1: Enter the function.
Step 2: Now select the variable “v”.
Step 3: Hit calculate button.
Step 4: The result will show below calculate button.
Step 5: Press show steps to get the step by step solution.
Wrap Up
In this post, the basics of derivative calculus have been covered with the help of solved examples. Now you can take assistance for evaluating the problems of differential calculus manually or using a calculator. We have suggested a calculator and solved examples for your assistance.