Question

What is the equation of the line tangent to `f(x)=(5+4x)^2` at `x=7`?

Answer 1

The slope of `f(x)=(5+4x)^2` at 7 is 264.

Explanation:

The derivative of a function gives the slope of a function at each point along that curve. Thus `{d f(x)}/dx` evaluated at x=a, is the slope of the function `f(x)`at `a`.

This function is
`f(x)=(5+4x)^2`, if you haven't learned the chain rule yet, you expand the polynomial to get `f(x)=25+ 40x + 16x^2`.

Using the fact that the derivative is linear, so constant multiplication and addition and subtraction is straightforward and then using derivative rule, `{d }/{dx} a x^n = n * a x^ {n-1}`, we get:
`{d f(x) }/dx=d/dx25+ d/dx40x + d/dx16x^2`

`{d f(x)}/{dx}=40 + 32x`.

This function gives the slope of `f(x)=(5+4x)^2` at any point, we are interested in the value at x=7 so we substitute 7 into the expression for the derivative.

`40 + 32(7)=264.`

Answer 2

y - 264x + 759 = 0

Explanation:

To find the equation of the tangent , y - b = m(x - a ) , require to find m and (a , b ) , a point on the line.

The derivative f'(7) will give the gradient of the tangent (m ) and evaluating f(7) will give (a , b ).

differentiate using the `color(blue)(" chain rule ")`

`f'(x) = 2(5+4x ) d/dx (5+4x ) = 8(5+ 4x )`

now f'(7) = 8(5+28) = 264and f(7) = `(5 + 28 )^2 = 1089`

now have m= 264 and (a , b ) = ( 7 , 1089 )

equation of tangent : y - 1089 = 264 (x - 7 )

hence y -1089 = 264x - 1848

`rArr y - 264x +759 = 0`