The following table lists the values of functions $${g}$$ and $${h}$$, and of their derivatives, $${g}^{\prime}$$ and $${h}^{\prime}$$, for the $${x}$$-values $$-2$$ and $$3$$.

![](http://admin.getprephub.com/media/editor/d115_20221016183226062758.png)
Evaluate $$\frac{d}{d x}[g(h(x))]$$ at $$x=-2$$



<div id="id_cont_display" class="markdown-body editormd-html-preview"><textarea>The following table lists the values of functions $${g}$$ and $${h}$$, and of their derivatives, $${g}^{\prime}$$ and $${h}^{\prime}$$, for the $${x}$$-values $$-2$$ and $$3$$.

![](http://admin.getprephub.com/media/editor/d115_20221016183226062758.png)
Evaluate $$\frac{d}{d x}[g(h(x))]$$ at $$x=-2$$

</textarea>The following table lists the values of functions <math><semantics><mrow><mrow><mi>g</mi></mrow></mrow><annotation encoding="application/x-tex">{g}</annotation></semantics></math>g and <math><semantics><mrow><mrow><mi>h</mi></mrow></mrow><annotation encoding="application/x-tex">{h}</annotation></semantics></math>h, and of their derivatives, <math><semantics><mrow><msup><mrow><mi>g</mi></mrow><mrow><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">{g}^{\prime}</annotation></semantics></math>g​′​​ and <math><semantics><mrow><msup><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">{h}^{\prime}</annotation></semantics></math>h​′​​, for the <math><semantics><mrow><mrow><mi>x</mi></mrow></mrow><annotation encoding="application/x-tex">{x}</annotation></semantics></math>x-values <math><semantics><mrow><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">-2</annotation></semantics></math>−2 and <math><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>3.
<img src="http://admin.getprephub.com/media/editor/d115_20221016183226062758.png" alt=""> Evaluate <math><semantics><mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>[</mo><mi>g</mi><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>]</mo></mrow><annotation encoding="application/x-tex">\frac{d}{d x}[g(h(x))]</annotation></semantics></math>​dx​​d​​[g(h(x))] at <math><semantics><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">x=-2</annotation></semantics></math>x=−2
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<div id="id_solution_display" class="markdown-body editormd-html-preview"><textarea>Given that,
$$\frac{d}{d x}[g(h(x))]=g^{\prime}(h(x)) h^{\prime}(x) $$
$$ \frac{d}{d x}[g(h(-2)]=g^{\prime}(h(-2)) h^{\prime}(-2). $$
$$ =g^{\prime}(3)\cdot(-5) $$
$$ =1 \cdot(-5) $$
$$ =-5 $$
</textarea>Given that, <math><semantics><mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>[</mo><mi>g</mi><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>]</mo><mo>=</mo><msup><mi>g</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><msup><mi>h</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\frac{d}{d x}[g(h(x))]=g^{\prime}(h(x)) h^{\prime}(x) </annotation></semantics></math>​dx​​d​​[g(h(x))]=g​′​​(h(x))h​′​​(x) <math><semantics><mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>[</mo><mi>g</mi><mo>(</mo><mi>h</mi><mo>(</mo><mo>−</mo><mn>2</mn><mo>)</mo><mo>]</mo><mo>=</mo><msup><mi>g</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>h</mi><mo>(</mo><mo>−</mo><mn>2</mn><mo>)</mo><mo>)</mo><msup><mi>h</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mo>−</mo><mn>2</mn><mo>)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex"> \frac{d}{d x}[g(h(-2)]=g^{\prime}(h(-2)) h^{\prime}(-2). </annotation></semantics></math>​dx​​d​​[g(h(−2)]=g​′​​(h(−2))h​′​​(−2). <math><semantics><mrow><mo>=</mo><msup><mi>g</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mn>3</mn><mo>)</mo><mo>⋅</mo><mo>(</mo><mo>−</mo><mn>5</mn><mo>)</mo></mrow><annotation encoding="application/x-tex"> =g^{\prime}(3)\cdot(-5) </annotation></semantics></math>=g​′​​(3)⋅(−5) <math><semantics><mrow><mo>=</mo><mn>1</mn><mo>⋅</mo><mo>(</mo><mo>−</mo><mn>5</mn><mo>)</mo></mrow><annotation encoding="application/x-tex"> =1 \cdot(-5) </annotation></semantics></math>=1⋅(−5) <math><semantics><mrow><mo>=</mo><mo>−</mo><mn>5</mn></mrow><annotation encoding="application/x-tex"> =-5 </annotation></semantics></math>=−5
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