In calculus, the derivative plays a vital role in finding the slope of the tangent line. Differential calculus is usually used to determine the derivative of the single, double, and multivariable functions. It can be solved either by using the laws of derivatives or the first principle method.
In the first principle method, limits are used to differentiate the functions. The limit is a branch of calculus used to define continuity, integration, and differentiation. In this article, we’ll study the basics of differential calculus along with its rules and examples.
What is the derivative?
In mathematics, the instant rate of change of a function with respect to its independent variable is known as differential calculus. The differential calculus is also used to find the tangent line’s slope.
The problems of calculus and differential equations can be solved by using derivatives. The problems of calculus and differential equations can be involved constant, exponential, linear, quadric, polynomial, logarithmic, and trigonometric functions.
The derivative function can be a single variable, double variable, equation, or multivariable. To solve the problems of single variable functions, explicit differentiation is used. It can be involved with several variables line x, y, z, u, v, w, etc.
It is denoted by d/dx, d/dy, d/dz, d/dv, d/dw, etc. the problems of double variable or equation functions can be solved by using implicit differentiation. In implicit derivatives, the function must contain two variables. It differentiates the dependent variable with respect to an independent variable.
It is dented by dy/dx and the expressions of y didn’t consider as constant while taking the derivative with respect to x. for example 2y3 gives 6y2 after differentiating with respect to x. for solving multivariable problems, the partial derivative is used.
It takes the multivariable functions and differentiates them with respect to any variable one by one. It is denoted by ∂/∂x, ∂/∂y, ∂/∂z, ∂/∂w, ∂/∂t, etc.
Rules of differentiation in calculus
In calculus, the differential problems can be solved easily either by using the rules of derivative or a derivative calculator. There are several rules of differentiation. Let’s study some of them briefly.
1. Constant rule of differentiation
If the given function is constant, then use the constant rule of differentiation. This rule states that the differential of each constant number and variable is always zero.
d/dw [F] = 0, where F is any constant
Example
Calculate the differential of 12y with respect to “w”.
Solution
Step-I: First of all, apply the differentiation notation on the given function.
d/dw [12y]
Step-II: Now calculate the above differential function.
As it is clear that the term 12y is a constant with respect to “w”.
d/dw [12y] = 0 (by constant rule of differentiation)
2. Power rule of differentiation
The power rule of differentiation states that the derivative of the exponential function is to multiply the power with the coefficient of the function and subtract 1 from the power of the function and find the differential of the function without power.
d/dw [pn(w)] = n * pn-1(w) * d/dw (p(w))
Example of power rule
Calculate the differential of 6w3 with respect to “w”.
Solution
Step-I: First of all, apply the differentiation notation on the given function.
d/dw [6w3]
Step-II: Now calculate the above differential function.
d/dw [6w3] = (6 * 3) w3 – 1 d/dw(w) … (by power rule of differentiation)
= (18) * w2 d/dw(w)
= (18) * w2 (1)
= (18) * w2
= 18w2
3. Sum rule of differentiation
The sum rule of differentiation states that the derivative notation must be applied to each function separately.
The general expressions of this rule of differentiation is:
- d/dw [p(w) + q(w)] = d/dw [p(w)] = d/dw [q(w)]
- d/dw [p(w) + q(w) + r(w)] = d/dw [p(w)] + d/dw [q(w)] + d/dw [r(w)]
Example of the sum rule
Calculate the differential of 5w5 + 4w2 + 2w with respect to “w”.
Solution
Step-I: First of all, apply the differentiation notation on the given function
Function = 5w5 + 4w2 + 2w
d/dw [p(w) + q(w) + r(w)] = d/dw [5w5 + 4w2 + 2w]
Step-II: By using the sum rule of derivative apply the notation of differentiations to each function separately.
d/dw [5w5 + 4w2 + 2w] = d/dw [5w5] + d/dw [4w2] + d/dw [2w]
Step-III: Now calculate the above differential function.
d/dw [5w5 + 4w2 + 2w] = 5 d/dw [w5] + 4 d/dw [w2] + 2 d/dw [w]
= 5 [5 w5 - 1] + 4 [2 w2 – 1] + 2 [w1 – 1]
= 5 [5 w4] + 4 [2 w1] + 2 [w0]
= 5 [5w4] + 4 [2w] + 2 [1]
= 5 * 5w4 + 4 * 2w + 2 [1]
= 25w4 + 8w + 2
4. Product rule of differentiation
The product rule of differentiation states that the differential of the multiplied functions must be taking the derivative of the first function with respect to the independent variable while the second function remains unchanged.
Plus, take the derivative of the second function with respect to the independent variable while the first function remains unchanged.
The general expression of the product rule of differentiation is:
- d/dw [p(w) * q(w)] = q(w) d/dw [p(w)] + p(w) d/dw [q(w)]
- d/dw [p(w) * q(w) * r(w)] = [d/dw (p(w)) * q(w) * r(w)] + [p(w) * d/dw (q(w)) * r(w)] + [p(w) * q(w) * d/dw (r(w)]
Example of the product rule
Calculate the differential of 4w3 * 2w2 * 3w with respect to “w”.
Solution
Step-I: First of all, apply the differentiation notation on the given function
Function = 4w3 * 2w2 * 3w
d/dw [p(w) * q(w) * r(w)] = d/dw [4w3 * 2w2 * 3w]
Step-II: By using the product rule of differentiation apply the notation of differentiations to each function one by one.
d/dw [p(w) * q(w) * r(w)] = [d/dw (p(w)) * q(w) * r(w)] + [p(w) * d/dw (q(w)) * r(w)] + [p(w) * q(w) * d/dw (r(w)]
= [d/dw (4w3) * 2w2 * 3w] + [4w3 * d/dw (2w2) * 3w] + [4w3 * 2w2 * d/dw (3w)]
Step-III: Now calculate the above differential function.
d/dw [4w3 * 2w2 * 3w] = [(4 x 3) w3 - 1) * 2w2 * 3w] + [4w3 * (2 x 2) w2 - 1) * 3w] + [4w3 * 2w2 * (3w1 - 1)]
= [(12) w2) * 2w2 * 3w] + [4w3 * (4) w1) * 3w] + [4w3 * 2w2 * (3w0)]
= [(12) w2) * 2w2 * 3w] + [4w3 * (4) w) * 3w] + [4w3 * 2w2 * (3(1))]
= [(12w2) * 2w2 * 3w] + [4w3 * 4w * 3w] + [4w3 * 2w2 * 3]
= [24w4 * 3w] + [16w4 * 3w] + [8w5 * 3]
= [72w5] + [48w5] + [24w5]
= 120w5 + [24w5]
= 144w5
Summary
Now after reading the above post, you can solve any problem of calculus and differential equations easily with the help of rules of differentiation. You can grab all the basics of derivatives just by learning the basics of this post.
