4 Simple Ways To Learn Derivatives In Calculus
Calculus is the branch of mathematics and the derivative is the core of calculus. Derivative can be defined in two ways, one is at a slope of a curve and the other is as a rate of change. Derivative can be used to describe the characteristic of the graph. Derivative is the challenging subject for most of the students. The derivative is the major part of mathematics and physics.
• Most important thing is first understand what is derivative?, why it is needed?
• Understand the derivative notation. There are two most important derivative notation.
a)Leibniz Notation
b)Lagrange’s Notation
• Memorize all important formula. Start from basics.
Here in this article we have 4 easy ways to solve derivative.
1: Explicit Differentiation
Explicit method is used when the equation is in the form of y=f(x), where f(x) can be x2, x5, x7 etc.
• If you have y=x2
then dy/dx= 2x
Here the slope of any tangent graph y=x2 is 2x.
• If you have y = 5
then dy/dx= 0
derivative of any constant is always zero.
• If you have y= 3x2+5x
then dy/dx = d(3x2)/dx+d(5x)/dx
dy/dx= 6x+ 5
• If you have y= x3(x+1) then make use of the product rule.
• If you have y= x4/(x-5) then make use of the quotient rule.
2: Implicit Differentiation
Implicit method is used when the equation cannot be written easily in y term.
Example: x3y+3y2=3x+5y
Let y=f(x)
=>x3 f(x)+3f(x)2=3x+5f(x)
Differentiate both side of the function with respect to x, using product rule
=>x3 f’(x)+3x2f(x)+6f(x)f’(x)=3+5f’(x)
Substitute f(x)=y
=>x3 f’(x)+3x2y+6yf’(x)=3+5f’(x)
=>x3 f’(x)+6yf’(x)-5f’(x)=3-3x2y
=>f’(x)(x3+6y-5)=(3-3x2y)
=>f’(x)=(3-3x2y)/(x3+6y-5)
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3: High Order Derivatives
Taking higher order derivative means just take derivative of the derivative equation. Take the second derivative of the derivative equation. Then take the third derivative of second derivative.
Example: If(x)= 6x3+5x2+3x+9
First derivative with respect to x
=>f’(x)=18x2+10x+3
Second derivative with respect to x
=>f’‘(x)=36x+10
Third derivative with respect to x
=>f’“(x)=36
4: The Chain Rule
When two or more function are given then make use of chain rule. Chain rule is a formula for computing the derivative of two or more composite function. Let f and g are two function then and F(x)=(fog)(x) then the derivative of F(x) is,
F’(x)=g’(f(x))f’(x)
Example: F(x)=(3x+1)3
Let f(x)=(3x+1)
And g(x)=x3
f’(x)=3
And g’(x)=3x2
Use chain rule F’(x)=g’(f(x))f’(x)
=>F’(x)=g’(3x+1)3
=>F’(x)=3(3x+1)2*3
=>F’(x)=9(3x+1)2
Tips:
• Practice as much as the problem you can. Practice problem base on explicit, implicit, high order derivation and chain rule.
• If you get a huge problem, don’t give up try to solve derivative problems using the product rule, quotient rule etc.
• Divide derivative problem in parts then solve it.
• Many websites and Free Math Tutoring Online provide step by step solution. Take math help from online.