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Differentiate 5 sin ⁡ ( 𝑥 3 + 2 𝑥 )

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The question asks to differentiate the function 5 sin(x^3 + 2x). First, recognize this as a composite function where the outer function is sin(u) and the inner function is u = x^3 + 2x. Using the chain rule, d/dx [sin(u)] = cos(u) * du/d......Login to view full explanation

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<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>The differentiable functions </mtext><mi>f</mi><mtext> and </mtext><mi>g</mi><mtext> are defined for all real numbers </mtext><mi>x</mi><mo>.</mo><mtext> Values of </mtext><mi>f</mi><mo>,</mo><msup><mi>f</mi><mo>′</mo></msup><mo>,</mo><mi>g</mi><mo>,</mo><mtext> and </mtext><msup><mi>g</mi><mo>′</mo></msup><mtext> for various values of </mtext><mi>x</mi><mtext> are given in the table.</mtext></math>\text{The differentiable functions } f \text{ and } g \text{ are defined for all real numbers } x. \text{ Values of } f, f', g, \text{ and } g' \text{ for various values of } x \text{ are given in the table.} <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Suppose that</mtext><mtext> </mtext><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext><mtext>Determine the following quantities.</mtext></math>\text{Suppose that} \ h(x)=f(g(x)) \ \text{and} \ H(x)=f(g(x)). \ \text{Determine the following quantities.} (a) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>h</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo></math>h'(1)=[Fill in the blank], (b) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>H</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo></math>H'(1)=[Fill in the blank],

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