Find the Difference Quotient f(x)=5x^2

Problem

ƒ(x)=5*x2

Solution

1. Identify the formula for the difference quotient, which is defined as the average rate of change over an interval h

(ƒ*(x+h)−ƒ(x))/h

2. Evaluate the function at x+h by substituting x+h into the original expression for every instance of x

ƒ*(x+h)=5*(x+h)2

3. Expand the squared binomial using the identity (a+b)2=a2+2*a*b+b2

ƒ*(x+h)=5*(x2+2*x*h+h2)

4. Distribute the constant factor 5 into the terms inside the parentheses.

ƒ*(x+h)=5*x2+10*x*h+5*h2

5. Substitute the expressions for ƒ*(x+h) and ƒ(x) into the difference quotient formula.

((5*x2+10*x*h+5*h2)−(5*x2))/h

6. Simplify the numerator by subtracting 5*x2 from 5*x2 which cancels those terms.

(10*x*h+5*h2)/h

7. Factor out the common term h from the numerator to prepare for division.

(h*(10*x+5*h))/h

8. Divide the numerator and denominator by h to reach the final simplified form.

10*x+5*h

Final Answer

(ƒ*(x+h)−ƒ(x))/h=10*x+5*h

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