Find the derivatives of: [latex]\mathrm{(a)}f(x)=10^{-x};\mathrm{and}\left({\mathrm{b}}\right)g(x)=x2^{3x}.[/latex]

(a) Rewrite [latex]f (x) = 10^{−x} = 10^{u},[/latex] where u = −x. Using (6.10.3) and the chain rule gives [latex]y=a^{x}\Rightarrow y^{\prime}=a^{x}\ln a[/latex] (6.10.3) [latex]f^{\prime}(x) = −10^{−x} \ln 10.[/latex] (b) Rewrite [latex]y = 2^{3x} = 2^{u},[/latex] where u = 3x. By (6.10.3) and the chain rule, [latex]y^{\prime} = (2^{u} \ln 2)u^{\prime} = (2^{3x} \ln 2) · 3 = 3 · 2^{3x} \ln 2[/latex] Finally, using the product rule we obtain [latex]g^{\prime}(x) = 1 · 2^{3x} + x · 3 · 2^{3x} \ln 2 = 2^{3x}(1 + 3x \ln 2)[/latex]

Question 6.10.4: Find the derivatives of: (a) f (x) = 10^−x; and (b) g(x) = x......

Essential Mathematics for Economic Analysis [1185377]

Find the derivatives of: $\mathrm{(a)}f(x)=10^{-x};\mathrm{and}\left({\mathrm{b}}\right)g(x)=x2^{3x}.$

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(a) Rewrite $f (x) = 10^{−x} = 10^{u},$ where u = −x. Using (6.10.3) and the chain rule gives

$y=a^{x}\Rightarrow y^{\prime}=a^{x}\ln a$ (6.10.3)

$f^{\prime}(x) = −10^{−x} \ln 10.$
(b) Rewrite $y = 2^{3x} = 2^{u},$ where u = 3x. By (6.10.3) and the chain rule,
$y^{\prime} = (2^{u} \ln 2)u^{\prime} = (2^{3x} \ln 2) · 3 = 3 · 2^{3x} \ln 2$
Finally, using the product rule we obtain
$g^{\prime}(x) = 1 · 2^{3x} + x · 3 · 2^{3x} \ln 2 = 2^{3x}(1 + 3x \ln 2)$