How do you find f'(x) using the definition of a derivative for #f(x)=(16t)/(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 onesided 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 onesided 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 onesided 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 onesided 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)=(x1)(x+2)^2 # at # x=1#?
 What is the equation of the line tangent to #f(x)=1/(x1)# at #x=2#?
 What is the equation of the line normal to # f(x)=(xsinx)/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^2sinx # at # x=pi/3#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7