Chapter 3 | Derivatives
253
Find the equation of the line tangent to the graph of f ( x ) = x 2 −4 x +6 at x =1.
Solution To find the equation of the tangent line, we need a point and a slope. To find the point, compute f (1) =1 2 −4(1)+6=3. This gives us the point (1, 3). Since the slope of the tangent line at 1 is f ′(1), we must first find f ′( x ). Using the definition of a derivative, we have f ′( x ) =2 x −4 so the slope of the tangent line is f ′(1) =−2. Using the point-slope formula, we see that the equation of the tangent line is y −3=−2( x −1). Putting the equation of the line in slope-intercept form, we obtain y =−2 x +5.
3.15 Find the equation of the line tangent to the graph of f ( x ) =3 x 2 −11 at x =2. Use the point-slope form.
The Product Rule Now that we have examined the basic rules, we can begin looking at some of the more advanced rules. The first one examines the derivative of the product of two functions. Although it might be tempting to assume that the derivative of the product is the product of the derivatives, similar to the sum and difference rules, the product rule does not follow this pattern. To see why we cannot use this pattern, consider the function f ( x ) = x 2 , whose derivative is f ′( x ) =2 x and not d dx ( x ) · d dx ( x ) =1·1=1.
Theorem 3.5: Product Rule Let f ( x ) and g ( x ) be differentiable functions. Then d dx ⎛ ⎝ f ( x ) g ( x ) ⎞ ⎠ = d dx ⎛ ⎝ f ( x ) ⎞
⎠ · g ( x )+ d dx ⎛
⎞ ⎠ · f ( x ).
⎝ g ( x )
That is,
if j ( x ) = f ( x ) g ( x ), then j ′( x ) = f ′( x ) g ( x )+ g ′( x ) f ( x ). This means that the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.
Proof We begin by assuming that f ( x ) and g ( x ) are differentiable functions. At a key point in this proof we need to use the fact that, since g ( x ) is differentiable, it is also continuous. In particular, we use the fact that since g ( x ) is continuous, lim h →0 g ( x + h ) = g ( x ).
Made with FlippingBook - professional solution for displaying marketing and sales documents online