Question Statement

Find the first and second derivatives of the function:

$f(x)=(-5x+9{)}^2$

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Background and Explanation

This problem involves differentiating a composite function using the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. You should also recall the power rule for differentiation and that the derivative of a constant is zero.

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Solution

We are given the function $f(x)=(-5x+9{)}^2$. This is a composite function where the outer function is $u^2$ and the inner function is $u=-5x+9$.

Finding the First Derivative $f^{\prime }(x)$

To find $f^{\prime }(x)$, we apply the chain rule combined with the power rule:

$\begin{array}{rl}{f}^{\prime }(x)& =\frac{d}{dx}\left[(-5x+9{)}^2\right]\\ & =2(-5x+9{)}^{2-1}\cdot \frac{d}{dx}(-5x+9)\text{(Power rule + Chain rule)}\\ & =2(-5x+9)\cdot (-5+0)\text{(Differentiating the inner function)}\\ & =2(-5x+9)\cdot (-5)\\ & =-10(-5x+9)\\ & =50x-90\text{(Expanding the expression)}\end{array}$

So the first derivative is: $f^{\prime }(x)=50x-90$

Finding the Second Derivative $f^{\prime \prime }(x)$

Now we differentiate $f^{\prime }(x)$ with respect to $x$ to find the second derivative:

\begin{aligned} f''(x) &= \frac{d}{dx}\left[50x - 90\right] \\ &= 50\frac{d}{dx}(x) - \frac{d}{dx}(90) \quad \text{(Linearity of differentiation)} \\ &= 50 \cdot (1) - 0 \quad \text{(Power rule:\frac(x) = 1$,constantrule:$\frac(90) = 0)} \\ &= 50 \end{aligned}

Therefore, the second derivative is constant: $f^{\prime \prime }(x)=50$

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Key Formulas or Methods Used

• Chain Rule: $\frac{d}{dx}[g(h(x))]=g^{\prime }(h(x))\cdot h^{\prime }(x)$
• Power Rule: $\frac{d}{dx}[x^n]=n{x}^{n-1}$
• Constant Rule: $\frac{d}{dx}[c]=0$ for any constant $c$
• Constant Multiple Rule: $\frac{d}{dx}[cf(x)]=c\cdot f^{\prime }(x)$
• Sum/Difference Rule: $\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)$

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Summary of Steps

1. Identify the structure: Recognize $f(x)=(-5x+9{)}^2$ as a composite function with outer function $u^2$ and inner function $u=-5x+9$.
2. Apply chain rule for first derivative: Differentiate the outer function (keeping the inner function unchanged), then multiply by the derivative of the inner function: $2(-5x+9)\cdot (-5)$.
3. Simplify: Expand $-10(-5x+9)$ to get $50x-90$.
4. Differentiate again for second derivative: Differentiate $50x-90$ term by term to obtain $50$.