As we have seen throughout the examples in this section, it seldom happens that we are called on to apply just one differentiation rule to find the derivative of a given function. At this point, by combining the differentiation rules, we may find the derivatives of any polynomial or rational function. Later on we will encounter more complex combinations of differentiation rules.
A good rule of thumb to use when applying several rules is to apply the rules in reverse of the order in which we would evaluate the function.
For [latex]k(x)=3h(x)+x^2g(x)[/latex], find [latex]k^{\prime}(x)[/latex].
Finding this derivative requires the sum rule, the constant multiple rule, and the product rule.
[latex]\begin{array}{lllll}k^{\prime}(x) & =\frac{d}{dx}(3h(x)+x^2g(x))=\frac{d}{dx}(3h(x))+\frac{d}{dx}(x^2g(x)) & & & \text{Apply the sum rule.} \\ & =3\frac{d}{dx}(h(x))+(\frac{d}{dx}(x^2)g(x)+\frac{d}{dx}(g(x))x^2) & & & \begin{array}{l}\text{Apply the constant multiple rule to} \\ \text{differentiate} \, 3h(x) \, \text{and the product} \\ \text{rule to differentiate} \, x^2g(x). \end{array} \\ & =3h^{\prime}(x)+2xg(x)+g^{\prime}(x)x^2 & & & \end{array}[/latex]
For [latex]h(x)=\large \frac{2x^3k(x)}{3x+2}[/latex], find [latex]h^{\prime}(x)[/latex].
This procedure is typical for finding the derivative of a rational function.
Determine the values of [latex]x[/latex] for which [latex]f(x)=x^3-7x^2+8x+1[/latex] has a horizontal tangent line.
To find the values of [latex]x[/latex] for which [latex]f(x)[/latex] has a horizontal tangent line, we must solve [latex]f^{\prime}(x)=0[/latex]. Since
we must solve [latex](3x-2)(x-4)=0[/latex]. Thus we see that the function has horizontal tangent lines at [latex]x=\frac{2}{3}[/latex] and [latex]x=4[/latex] as shown in the following graph.
Figure 2. This function has horizontal tangent lines at [latex]x = 2/3[/latex] and [latex]x = 4[/latex].
Watch the following video to see the worked solution to this example.
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