Find The Derivative Of The Function Using The Definition.Given:${ Y = 7x^2 - 5x } F I N D : Find: F In D : { Y' = \}

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Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. There are several ways to find the derivative of a function, including using the definition of a derivative, the power rule, and the product rule. In this article, we will focus on finding the derivative of a function using the definition.

The Definition of a Derivative

The definition of a derivative is given by:

fβ€²(x)=lim⁑hβ†’0f(x+h)βˆ’f(x)h{ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} }

This definition states that the derivative of a function f(x) at a point x is equal to the limit of the difference quotient as h approaches zero.

Finding the Derivative of the Function

Given the function:

y=7x2βˆ’5x{ y = 7x^2 - 5x }

We want to find the derivative of this function using the definition.

Step 1: Substitute the Function into the Definition

Substituting the function y = 7x^2 - 5x into the definition of a derivative, we get:

yβ€²=lim⁑hβ†’0(7(x+h)2βˆ’5(x+h))βˆ’(7x2βˆ’5x)h{ y' = \lim_{h \to 0} \frac{(7(x + h)^2 - 5(x + h)) - (7x^2 - 5x)}{h} }

Step 2: Expand the Squared Term

Expanding the squared term in the numerator, we get:

yβ€²=lim⁑hβ†’0(7(x2+2hx+h2)βˆ’5xβˆ’5h)βˆ’(7x2βˆ’5x)h{ y' = \lim_{h \to 0} \frac{(7(x^2 + 2hx + h^2) - 5x - 5h) - (7x^2 - 5x)}{h} }

Step 3: Simplify the Numerator

Simplifying the numerator, we get:

yβ€²=lim⁑hβ†’07x2+14hx+7h2βˆ’5xβˆ’5hβˆ’7x2+5xh{ y' = \lim_{h \to 0} \frac{7x^2 + 14hx + 7h^2 - 5x - 5h - 7x^2 + 5x}{h} }

Step 4: Cancel Out the Like Terms

Canceling out the like terms in the numerator, we get:

yβ€²=lim⁑hβ†’014hx+7h2βˆ’5hh{ y' = \lim_{h \to 0} \frac{14hx + 7h^2 - 5h}{h} }

Step 5: Factor Out the Common Term

Factoring out the common term h in the numerator, we get:

yβ€²=lim⁑hβ†’0h(14x+7hβˆ’5)h{ y' = \lim_{h \to 0} \frac{h(14x + 7h - 5)}{h} }

Step 6: Cancel Out the Common Term

Canceling out the common term h in the numerator and denominator, we get:

yβ€²=lim⁑hβ†’0(14x+7hβˆ’5){ y' = \lim_{h \to 0} (14x + 7h - 5) }

Step 7: Evaluate the Limit

Evaluating the limit as h approaches zero, we get:

yβ€²=14xβˆ’5{ y' = 14x - 5 }

Conclusion

In this article, we used the definition of a derivative to find the derivative of the function y = 7x^2 - 5x. We substituted the function into the definition, expanded the squared term, simplified the numerator, canceled out the like terms, factored out the common term, canceled out the common term, and evaluated the limit to find the derivative.

Final Answer

The derivative of the function y = 7x^2 - 5x is:

yβ€²=14xβˆ’5{ y' = 14x - 5 }

Example Problems

Problem 1

Find the derivative of the function y = 3x^3 - 2x^2 + x.

Solution

Using the definition of a derivative, we get:

yβ€²=lim⁑hβ†’0(3(x+h)3βˆ’2(x+h)2+(x+h))βˆ’(3x3βˆ’2x2+x)h{ y' = \lim_{h \to 0} \frac{(3(x + h)^3 - 2(x + h)^2 + (x + h)) - (3x^3 - 2x^2 + x)}{h} }

Expanding the squared term in the numerator, we get:

yβ€²=lim⁑hβ†’0(3(x3+3hx2+3h2x+h3)βˆ’2(x2+2hx+h2)+x+h)βˆ’(3x3βˆ’2x2+x)h{ y' = \lim_{h \to 0} \frac{(3(x^3 + 3hx^2 + 3h^2x + h^3) - 2(x^2 + 2hx + h^2) + x + h) - (3x^3 - 2x^2 + x)}{h} }

Simplifying the numerator, we get:

yβ€²=lim⁑hβ†’03x3+9hx2+9h2x+3h3βˆ’2x2βˆ’4hxβˆ’2h2+x+hβˆ’3x3+2x2βˆ’xh{ y' = \lim_{h \to 0} \frac{3x^3 + 9hx^2 + 9h^2x + 3h^3 - 2x^2 - 4hx - 2h^2 + x + h - 3x^3 + 2x^2 - x}{h} }

Canceling out the like terms in the numerator, we get:

yβ€²=lim⁑hβ†’09hx2+9h2x+3h3βˆ’4hxβˆ’2h2+hh{ y' = \lim_{h \to 0} \frac{9hx^2 + 9h^2x + 3h^3 - 4hx - 2h^2 + h}{h} }

Factoring out the common term h in the numerator, we get:

yβ€²=lim⁑hβ†’0h(9x2+9hx+3h2βˆ’4xβˆ’2h+1)h{ y' = \lim_{h \to 0} \frac{h(9x^2 + 9hx + 3h^2 - 4x - 2h + 1)}{h} }

Canceling out the common term h in the numerator and denominator, we get:

yβ€²=lim⁑hβ†’0(9x2+9hx+3h2βˆ’4xβˆ’2h+1){ y' = \lim_{h \to 0} (9x^2 + 9hx + 3h^2 - 4x - 2h + 1) }

Evaluating the limit as h approaches zero, we get:

yβ€²=9x2βˆ’4x+1{ y' = 9x^2 - 4x + 1 }

Problem 2

Find the derivative of the function y = 2x^4 + 3x^2 - x.

Solution

Using the definition of a derivative, we get:

yβ€²=lim⁑hβ†’0(2(x+h)4+3(x+h)2βˆ’(x+h))βˆ’(2x4+3x2βˆ’x)h{ y' = \lim_{h \to 0} \frac{(2(x + h)^4 + 3(x + h)^2 - (x + h)) - (2x^4 + 3x^2 - x)}{h} }

Expanding the squared term in the numerator, we get:

yβ€²=lim⁑hβ†’0(2(x4+4hx3+6h2x2+4h3x+h4)+3(x2+2hx+h2)βˆ’xβˆ’h)βˆ’(2x4+3x2βˆ’x)h{ y' = \lim_{h \to 0} \frac{(2(x^4 + 4hx^3 + 6h^2x^2 + 4h^3x + h^4) + 3(x^2 + 2hx + h^2) - x - h) - (2x^4 + 3x^2 - x)}{h} }

Simplifying the numerator, we get:

yβ€²=lim⁑hβ†’02x4+8hx3+12h2x2+8h3x+2h4+3x2+6hx+3h2βˆ’xβˆ’hβˆ’2x4βˆ’3x2+xh{ y' = \lim_{h \to 0} \frac{2x^4 + 8hx^3 + 12h^2x^2 + 8h^3x + 2h^4 + 3x^2 + 6hx + 3h^2 - x - h - 2x^4 - 3x^2 + x}{h} }

Canceling out the like terms in the numerator, we get:

yβ€²=lim⁑hβ†’08hx3+12h2x2+8h3x+2h4+6hx+3h2βˆ’hh{ y' = \lim_{h \to 0} \frac{8hx^3 + 12h^2x^2 + 8h^3x + 2h^4 + 6hx + 3h^2 - h}{h} }

Factoring out the common term h in the numerator, we get:

yβ€²=lim⁑hβ†’0h(8x3+12hx+8h2x+2h3+6x+3hβˆ’1)h{ y' = \lim_{h \to 0} \frac{h(8x^3 + 12hx + 8h^2x + 2h^3 + 6x + 3h - 1)}{h} }

Canceling out the common term h in the numerator and denominator, we get:

yβ€²=lim⁑hβ†’0(8x3+12hx+8h2x+2h3+6x+3hβˆ’1){ y' = \lim_{h \to 0} (8x^3 + 12hx + 8h^2x + 2h^3 + 6x + 3h - 1) }

Evaluating the limit as h approaches zero, we get:

yβ€²=8x3+6xβˆ’1{ y' = 8x^3 + 6x - 1 }

Conclusion

In this article, we used the definition of a derivative to find the derivative of two functions. We substituted the functions into the definition, expanded the squared term, simplified the numerator, canceled out the like terms, factored out the common term, canceled out the common term, and evaluated the limit to find the derivative.

Final Answer

The derivative of the function y = 7x^2 - 5x is:

yβ€²=14xβˆ’5{ y' = 14x - 5 }

The derivative of the function y = 3x^3 - 2x^2 + x is:

yβ€²=9x2βˆ’4x+1{ y' = 9x^2 - 4x + 1 }

The derivative of the function y = 2x^4 + 3x^2 - x is:

yβ€²=8x3+6xβˆ’1{ y' = 8x^3 + 6x - 1 }

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Q: What is the definition of a derivative?

A: The definition of a derivative is given by:

fβ€²(x)=lim⁑hβ†’0f(x+h)βˆ’f(x)h{ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} }

This definition states that the derivative of a function f(x) at a point x is equal to the limit of the difference quotient as h approaches zero.

Q: How do I find the derivative of a function using the definition?

A: To find the derivative of a function using the definition, you need to follow these steps:

  1. Substitute the function into the definition of a derivative.
  2. Expand the squared term in the numerator.
  3. Simplify the numerator.
  4. Cancel out the like terms in the numerator.
  5. Factor out the common term in the numerator.
  6. Cancel out the common term in the numerator and denominator.
  7. Evaluate the limit as h approaches zero.

Q: What is the difference between the definition of a derivative and the power rule?

A: The definition of a derivative is a general formula for finding the derivative of a function, while the power rule is a specific rule for finding the derivative of a function with a power of x. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).

Q: Can I use the definition of a derivative to find the derivative of a function with a power of x?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a power of x. However, it may be more difficult and time-consuming than using the power rule.

Q: What is the significance of the limit in the definition of a derivative?

A: The limit in the definition of a derivative represents the rate of change of the function as the input approaches a specific value. It is an essential concept in calculus and is used to define the derivative of a function.

Q: Can I use the definition of a derivative to find the derivative of a function with multiple variables?

A: Yes, you can use the definition of a derivative to find the derivative of a function with multiple variables. However, it may be more difficult and time-consuming than using the chain rule or the partial derivative rule.

Q: What is the relationship between the derivative of a function and the function itself?

A: The derivative of a function represents the rate of change of the function with respect to its input. It is a measure of how fast the function changes as the input changes.

Q: Can I use the definition of a derivative to find the derivative of a function with a trigonometric function?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a trigonometric function. However, it may be more difficult and time-consuming than using the chain rule or the trigonometric derivative rule.

Q: What is the significance of the derivative of a function in real-world applications?

A: The derivative of a function is used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits. It is an essential tool in physics, engineering, and economics.

Q: Can I use the definition of a derivative to find the derivative of a function with a logarithmic function?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a logarithmic function. However, it may be more difficult and time-consuming than using the chain rule or the logarithmic derivative rule.

Q: What is the relationship between the derivative of a function and the function's graph?

A: The derivative of a function represents the slope of the function's graph at a specific point. It is a measure of how steep the graph is at that point.

Q: Can I use the definition of a derivative to find the derivative of a function with a polynomial function?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a polynomial function. However, it may be more difficult and time-consuming than using the power rule.

Q: What is the significance of the derivative of a function in optimization problems?

A: The derivative of a function is used to find the maximum or minimum of the function, which is essential in optimization problems.

Q: Can I use the definition of a derivative to find the derivative of a function with a rational function?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a rational function. However, it may be more difficult and time-consuming than using the quotient rule.

Q: What is the relationship between the derivative of a function and the function's inverse?

A: The derivative of a function's inverse is the reciprocal of the derivative of the function. This is known as the inverse function theorem.

Q: Can I use the definition of a derivative to find the derivative of a function with a piecewise function?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a piecewise function. However, it may be more difficult and time-consuming than using the piecewise derivative rule.

Q: What is the significance of the derivative of a function in physics and engineering?

A: The derivative of a function is used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits. It is an essential tool in physics and engineering.

Q: Can I use the definition of a derivative to find the derivative of a function with a vector-valued function?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a vector-valued function. However, it may be more difficult and time-consuming than using the chain rule or the vector derivative rule.

Q: What is the relationship between the derivative of a function and the function's integral?

A: The derivative of a function's integral is the function itself. This is known as the fundamental theorem of calculus.

Q: Can I use the definition of a derivative to find the derivative of a function with a parametric function?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a parametric function. However, it may be more difficult and time-consuming than using the chain rule or the parametric derivative rule.

Q: What is the significance of the derivative of a function in economics?

A: The derivative of a function is used to model real-world phenomena, such as the behavior of supply and demand, the growth of economies, and the behavior of financial markets. It is an essential tool in economics.

Q: Can I use the definition of a derivative to find the derivative of a function with a multivariable function?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a multivariable function. However, it may be more difficult and time-consuming than using the chain rule or the multivariable derivative rule.

Q: What is the relationship between the derivative of a function and the function's partial derivatives?

A: The derivative of a function's partial derivatives is the function itself. This is known as the fundamental theorem of calculus for multivariable functions.

Q: Can I use the definition of a derivative to find the derivative of a function with a function of several variables?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a function of several variables. However, it may be more difficult and time-consuming than using the chain rule or the function of several variables derivative rule.

Q: What is the significance of the derivative of a function in computer science?

A: The derivative of a function is used to model real-world phenomena, such as the behavior of algorithms, the growth of data, and the behavior of computer networks. It is an essential tool in computer science.

Q: Can I use the definition of a derivative to find the derivative of a function with a function of a complex variable?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a function of a complex variable. However, it may be more difficult and time-consuming than using the chain rule or the function of a complex variable derivative rule.

Q: What is the relationship between the derivative of a function and the function's complex derivative?

A: The derivative of a function's complex derivative is the function itself. This is known as the fundamental theorem of calculus for complex functions.

Q: Can I use the definition of a derivative to find the derivative of a function with a function of a matrix?

A: Yes, you can use the definition of a derivative to find the derivative of a function with a function of a matrix. However, it may be more difficult and time-consuming than using the chain rule or the function of a matrix derivative rule.

Q: What is the significance of the derivative of a function in machine learning?

A: The derivative of a function is used to model real-world phenomena, such as the behavior of neural networks, the growth of data, and the behavior of computer networks. It is an essential tool in machine learning.

**Q: Can I use the definition of a derivative