Sample Problems on Derivatives
Here we have provided you with some solved problems on Derivatives:
Question 1: Find the derivative of the function f(x) = x2 at x = 0 using the First Principle.
Solution:
f'(x) = [Tex]\lim_{h \to 0}\frac{f(x + h) – f(x)}{h} [/Tex]
f'(x) = [Tex]\lim_{h \to 0}\frac{(x + h)^2 – x^2}{h}[/Tex]
⇒ f'(x) = [Tex]\lim_{h \to 0}\frac{(x^2 + h^2 + 2hx – x^2)}{h}[/Tex]
⇒ f'(x) = [Tex]\lim_{h \to 0}h + 2x[/Tex]
⇒ f'(x) = 2x
f'(0) = 0
Question 2: Find the derivative of the function f(x) = x2 at x = 2 by Limit Definition.
Solution:
f'(x) = [Tex]\lim_{h \to 0}\frac{f(x + h) – f(x)}{h} [/Tex]
f'(x) = [Tex]\lim_{h \to 0}\frac{(x + h)^2 – x^2}{h}[/Tex]
⇒ f'(x) = [Tex]\lim_{h \to 0}\frac{(x^2 + h^2 + 2hx – x^2)}{h}[/Tex]
⇒ f'(x) = [Tex]\lim_{h \to 0}h + 2x[/Tex]
⇒ f'(x) = 2x
f'(2) = 4
Question 3: Find the derivative of the function f(x) = x2 + x +1 at x = 0.
Solution:
f'(x) = [Tex]\lim_{h \to 0}\frac{f(x + h) – f(x)}{h} [/Tex]
f'(x) = [Tex]\lim_{h \to 0}\frac{((x + h)^2 + (x +h) + 1) – (x^2 + x + 1)}{h}[/Tex]
⇒ f'(x) = [Tex]\lim_{h \to 0}\frac{((x^2 + h^2 + 2hx + (x +h) + 1) – (x^2 + x + 1)}{h}[/Tex]
⇒ f'(x) = [Tex]\lim_{h \to 0}\frac{(h^2 + 2hx +h)}{h}[/Tex]
⇒ f'(x) = [Tex]\lim_{h \to 0}(h + 2x +1)[/Tex]
⇒f'(x) = 2x + 1
f'(0) = 1
Question 4: Find the derivative of the function f(x) = ex at x = 0.
Solution:
f'(x) = [Tex]\lim_{h \to 0}\frac{f(x + h) – f(x)}{h} [/Tex]
f'(x) = [Tex]\lim_{h \to 0}\frac{e^{(x + h)} – e^x}{h}[/Tex]
⇒ f'(x) = [Tex]\lim_{h \to 0}\frac{e^xe^h – e^x}{h}[/Tex]
⇒ f'(x) = [Tex]e^x\lim_{h \to 0}\frac{(e^h – 1)}{h}[/Tex]
This is 0/0 form of the limit. We know that [Tex]\lim_{h \to 0}\frac{(e^h – 1)}{h} = 1[/Tex]
⇒ f'(x) = [Tex]e^x\lim_{h \to 0}\frac{e^h }{1}[/Tex]
⇒ f'(x) = [Tex]e^x (1)[/Tex]
⇒f'(x) =ex
f'(0) = 1
Notice that the derivative of exponential function is exponential itself.
Derivatives | First and Second Order Derivatives, Formulas and Examples
Derivatives: In mathematics, a Derivative represents the rate at which a function changes as its input changes. It measures how a function’s output value moves as its input value nudges a little bit. This concept is a fundamental piece of calculus. It is used extensively across science, engineering, economics, and more to analyze changes.
Derivative
Table of Content
- What are Derivatives?
- Derivatives Meaning
- Derivative by First Principle
- Types of Derivatives
- First Order Derivative
- Second Order Derivative
- nth Order Derivative
- Derivatives Formula
- Rules of Derivatives
- Derivative of Composite Function
- Chain Rule of Derivatives
- Derivative of Implicit Function
- Parametric Derivatives
- Higher Order Derivatives
- Partial Derivative
- Logarithmic Differentiation
- Applications of Derivatives
- Derivatives Examples
- Sample Problems on Derivatives
- Practice Problems on Derivatives
A derivative is a calculus tool that measures the sensitivity of a function’s output to its input. It is also known as the instantaneous rate of change of a function at a given position.
The derivative of a function with just one variable is the slope of the line that is tangent to the function’s graph for a given input value. In terms of geometry, the derivative of a function can be defined as the slope of its graph.
In this article we have covered Meaning of Derivatives along with types of derivatives, examples, formulas, applications and many more.
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