Find the Second Derivative f(x) = square root of -7x-5
The problem is asking for the calculation of the second derivative of the function f(x), which is the square root of the expression (-7x - 5). To find the second derivative, you would first need to determine the first derivative of f(x) by applying the rules of differentiation to the square root function and the inner function (-7x - 5), and then differentiate that result once more to obtain the second derivative. The process involves applying the chain rule, the power rule, and simplifying the resulting expressions. The second derivative gives information about the curvature or concavity of the function's graph.
$f \left(\right. x \left.\right) = \sqrt{- 7 x - 5}$
Answer
Solution:
Step:1
Find the first derivative.
Step:1.1
Rewrite the function $f(x) = \sqrt{-7x - 5}$ as $f(x) = (-7x - 5)^{\frac{1}{2}}$.
Step:1.2
Apply the chain rule for differentiation: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, where $f(x) = x^{\frac{1}{2}}$ and $g(x) = -7x - 5$.
Step:1.2.1
Let $u = -7x - 5$. Then, $f(x) = \frac{d}{du}(u^{\frac{1}{2}})\frac{d}{dx}(-7x - 5)$.
Step:1.2.2
Differentiate $u^{\frac{1}{2}}$ using the power rule: $\frac{d}{du}(u^n) = nu^{n-1}$, where $n = \frac{1}{2}$.
Step:1.2.3
Substitute $u$ back into the equation: $f(x) = \frac{1}{2}(-7x - 5)^{-\frac{1}{2}}\frac{d}{dx}(-7x - 5)$.
Step:1.3
Express $-1$ as a fraction with a common denominator: $f(x) = \frac{1}{2}(-7x - 5)^{-\frac{1}{2}}\frac{d}{dx}(-7x - 5)$.
Step:1.4
Combine the exponents: $f(x) = \frac{1}{2}(-7x - 5)^{-\frac{1}{2}}\frac{d}{dx}(-7x - 5)$.
Step:1.5
Simplify the exponent: $f(x) = \frac{1}{2}(-7x - 5)^{-\frac{1}{2}}\frac{d}{dx}(-7x - 5)$.
Step:1.6
Simplify further: $f(x) = \frac{1}{2}(-7x - 5)^{-\frac{1}{2}}\frac{d}{dx}(-7x - 5)$.
Step:1.7
Combine the fractions: $f(x) = \frac{1}{2(-7x - 5)^{\frac{1}{2}}}\frac{d}{dx}(-7x - 5)$.
Step:1.8
Differentiate $-7x - 5$ using the sum rule: $f(x) = \frac{1}{2(-7x - 5)^{\frac{1}{2}}}(-7\frac{d}{dx}x + \frac{d}{dx}(-5))$.
Step:1.9
Differentiate $-7x$ and $-5$: $f(x) = \frac{1}{2(-7x - 5)^{\frac{1}{2}}}(-7 + 0)$.
Step:1.10
Simplify the derivative: $f(x) = -\frac{7}{2(-7x - 5)^{\frac{1}{2}}}$.
Step:2
Find the second derivative.
Step:2.1
Differentiate the first derivative using the constant multiple rule: $f''(x) = -\frac{7}{2}\frac{d}{dx}((-7x - 5)^{-\frac{1}{2}})$.
Step:2.2
Apply the chain rule again: $f''(x) = -\frac{7}{2}(-\frac{1}{2}(-7x - 5)^{-\frac{3}{2}}\frac{d}{dx}(-7x - 5))$.
Step:2.3
Simplify the expression: $f''(x) = \frac{7}{4}(-7x - 5)^{-\frac{3}{2}}(-7)$.
Step:2.4
Complete the differentiation: $f''(x) = -\frac{49}{4(-7x - 5)^{\frac{3}{2}}}$.
Step:3
The second derivative of $f(x)$ with respect to $x$ is $-\frac{49}{4(-7x - 5)^{\frac{3}{2}}}$.
Knowledge Notes:
Square Root as a Power: The square root of a function can be expressed as the function raised to the power of $\frac{1}{2}$.
Chain Rule: A fundamental differentiation rule used when finding the derivative of a composite function, expressed as $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$.
Power Rule: A differentiation rule that states $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n$ is a real number.
Sum Rule: A rule for differentiation that allows us to take the derivative of a sum of functions individually, expressed as $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$.
Constant Multiple Rule: This rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function, expressed as $\frac{d}{dx}[cf(x)] = cf'(x)$.
Negative Exponent Rule: For any nonzero number $b$ and any integer $n$, $b^{-n} = \frac{1}{b^n}$.
Differentiating Constants: The derivative of a constant is zero.
Combining Fractions: When combining fractions with common denominators, the numerators are added or subtracted while the denominator remains the same.
