How do you find f'(x) using the definition of a derivative for #f(x)=(1-6t)/(5+t)#?
By signing up, you agree to our Terms of Service and Privacy Policy
To find ( f'(x) ) using the definition of a derivative for ( f(x) = \frac{{1 - 6x}}{{5 + x}} ), follow these steps:
-
Start with the definition of the derivative: [ f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} ]
-
Substitute the function ( f(x) ): [ f'(x) = \lim_{{h \to 0}} \frac{{\frac{{1 - 6(x + h)}}{{5 + (x + h)}} - \frac{{1 - 6x}}{{5 + x}}}}{h} ]
-
Simplify the expression: [ f'(x) = \lim_{{h \to 0}} \frac{{\frac{{1 - 6x - 6h}}{{5 + x + h}} - \frac{{1 - 6x}}{{5 + x}}}}{h} ] [ f'(x) = \lim_{{h \to 0}} \frac{{\frac{{1 - 6x - 6h}}{{5 + x + h}} - \frac{{(1 - 6x)(5 + h)}}{{(5 + x)(5 + h)}}}}{h} ]
-
Combine fractions and simplify further: [ f'(x) = \lim_{{h \to 0}} \frac{{(1 - 6x - 6h)(5 + x) - (1 - 6x)(5 + h)}}{{h(5 + x)(5 + h)}} ]
-
Expand and simplify the numerator: [ f'(x) = \lim_{{h \to 0}} \frac{{5 - 30x - 30h + x - 6x^2 - 6xh - 5 - h - 30x + 6x^2 + 5 - 6xh}}{{h(5 + x)(5 + h)}} ] [ f'(x) = \lim_{{h \to 0}} \frac{{-31h}}{{h(5 + x)(5 + h)}} ]
-
Cancel out ( h ) from the numerator and denominator: [ f'(x) = \lim_{{h \to 0}} \frac{{-31}}{{(5 + x)(5 + h)}} ]
-
Evaluate the limit as ( h ) approaches 0: [ f'(x) = \frac{{-31}}{{(5 + x)(5)}} ] [ f'(x) = \frac{{-31}}{{(5 + x)(5)}} ]
So, the derivative of ( f(x) ) with respect to ( x ) is ( f'(x) = \frac{{-31}}{{(5 + x)(5)}} ).
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.
- What is the equation of the line normal to # f(x)=(x-1)(x+2)^2 # at # x=1#?
- What is the equation of the line tangent to #f(x)=1/(x-1)# at #x=2#?
- What is the equation of the line normal to # f(x)=(x-sinx)/tanx# at # x=pi/3#?
- What is the equation of the line normal to # f(x)=e^(sqrtx/x)# at # x=4#?
- What is the equation of the line that is normal to #f(x)=-3x^2-sinx # at # x=pi/3#?

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