Question

exercsies 3.3 differentiation rules
score: 2/40 answered: 1/19
question 2
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find the derivative of ( f(x)=8 x^{7}+10 x^{5} ).
( f^{prime}(x)= )
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Explanation:

Step1: Apply the sum rule

The sum rule states that if \(f(x)=u(x)+v(x)\), then \(f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x)\). Here \(u(x) = 8x^{7}\) and \(v(x)=10x^{5}\), so \(f^{\prime}(x)=\frac{d}{dx}(8x^{7})+\frac{d}{dx}(10x^{5})\).

Step2: Apply the constant - multiple rule

The constant - multiple rule states that if \(y = cf(x)\), then \(y^{\prime}=cf^{\prime}(x)\). For \(y = 8x^{7}\), \(c = 8\) and \(f(x)=x^{7}\), so \(\frac{d}{dx}(8x^{7})=8\frac{d}{dx}(x^{7})\); for \(y = 10x^{5}\), \(c = 10\) and \(f(x)=x^{5}\), so \(\frac{d}{dx}(10x^{5})=10\frac{d}{dx}(x^{5})\).

Step3: Apply the power rule

The power rule states that if \(y = x^{n}\), then \(y^{\prime}=nx^{n - 1}\).
For \(y=x^{7}\), \(n = 7\), so \(\frac{d}{dx}(x^{7})=7x^{6}\); for \(y=x^{5}\), \(n = 5\), so \(\frac{d}{dx}(x^{5})=5x^{4}\).
Substitute back:
\(\frac{d}{dx}(8x^{7})=8\times7x^{6}=56x^{6}\) and \(\frac{d}{dx}(10x^{5})=10\times5x^{4}=50x^{4}\).

Answer:

\(f^{\prime}(x)=56x^{6}+50x^{4}\)