How do you find the derivative using the difference quotient #f(x) = 5 / (x^2 + 7)#?
The quotient rule is:
So:
By signing up, you agree to our Terms of Service and Privacy Policy
The details depend on which difference quotient you use :
or,
By signing up, you agree to our Terms of Service and Privacy Policy
To find the derivative using the difference quotient for ( f(x) = \frac{5}{x^2 + 7} ), first, apply the definition of the difference quotient. Then simplify the expression and take the limit as ( h ) approaches ( 0 ). This will give you the derivative of ( f(x) ) with respect to ( x ).
By signing up, you agree to our Terms of Service and Privacy Policy
To find the derivative of ( f(x) = \frac{5}{x^2 + 7} ) using the difference quotient, we start by using the definition of the derivative:
[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]
Substitute the function ( f(x) = \frac{5}{x^2 + 7} ) into the difference quotient:
[ f'(x) = \lim_{h \to 0} \frac{\frac{5}{(x + h)^2 + 7} - \frac{5}{x^2 + 7}}{h} ]
Now, let's simplify this expression:
[ f'(x) = \lim_{h \to 0} \frac{5(x^2 + 7) - 5((x + h)^2 + 7)}{h((x^2 + 7)((x + h)^2 + 7))} ]
[ f'(x) = \lim_{h \to 0} \frac{5x^2 + 35 - (5x^2 + 10hx + 5h^2 + 35)}{h((x^2 + 7)((x + h)^2 + 7))} ]
[ f'(x) = \lim_{h \to 0} \frac{-10hx - 5h^2}{h((x^2 + 7)((x + h)^2 + 7))} ]
Now, we can cancel out ( h ) from the numerator and denominator:
[ f'(x) = \lim_{h \to 0} \frac{-10x - 5h}{(x^2 + 7)((x + h)^2 + 7)} ]
[ f'(x) = \frac{-10x}{(x^2 + 7)^2} ]
Therefore, the derivative of ( f(x) = \frac{5}{x^2 + 7} ) using the difference quotient is ( f'(x) = \frac{-10x}{(x^2 + 7)^2} ).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- How do you find the derivative of #2/(x+1)#?
- How do you use the chain rule to differentiate #y=8(x^4-x+1)^(3/4)#?
- What is the slope of the tangent line of #(x-y)^3e^y= C #, where C is an arbitrary constant, at #(-2,1)#?
- What is the derivative of #-4/x^2#?
- What is the implicit derivative of #3=(1-y)/x^2+xy #?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7