Question

differentiate the function.
$h(x) = (x - 5)(5x + 15)$
$h(x) = \square$
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differentiate the function.
$g(t) = 2t^{-3/4}$
$g(t) = \square$
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Explanation:

Step1: Differentiate \(h(x)=(x-5)(5x+15)\)

Use product rule: if \(h(x)=f(x)g(x)\), \(h'(x)=f'(x)g(x)+f(x)g'(x)\). Let \(f(x)=x-5\), \(g(x)=5x+15\). Then \(f'(x)=1\), \(g'(x)=5\). So \(h'(x)=1(5x+15)+(x-5)5=5x+15+5x-25=10x-10\).

Step2: Differentiate \(g(t)=2t^{-3/4}\)

Use power rule: \(\frac{d}{dt}(t^n)=nt^{n-1}\). So \(g'(t)=2*(-\frac{3}{4})t^{-3/4-1}=-\frac{3}{2}t^{-7/4}\).

Answer:

\(h'(x)=10x - 10\)
\(g'(t)=-\frac{3}{2}t^{-7/4}\)