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Magazine Chapter 3: Problem 12
Find \(d y / d x.\) $$y=7 x^{-6}-5 \sqrt{x}$$
Short Answer
Expert verified
\(\frac{dy}{dx} = -42x^{-7} - \frac{5}{2}x^{-1/2}\)
Step by step solution
01
Rewrite the expression with powers
First, rewrite the function for clarity by expressing all terms using exponents. \( y = 7x^{-6} - 5\sqrt{x} \) becomes \( y = 7x^{-6} - 5x^{1/2} \).
02
Apply the power rule for differentiation
Differentiating each term separately, recall that the derivative of \( x^n \) is \( nx^{n-1} \).
03
Differentiate the first term
For the term \( 7x^{-6} \), apply the power rule: the derivative is \(-42x^{-7}\).
04
Differentiate the second term
For the term \(-5x^{1/2}\), apply the power rule: the derivative is \(-\frac{5}{2}x^{-1/2}\).
05
Combine derivatives
Combine the derivatives of each term to form the derivative of the function: \( \frac{dy}{dx} = -42x^{-7} - \frac{5}{2}x^{-1/2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Rule
In calculus, the power rule is a fundamental tool used when finding the derivative of functions involving powers of a variable, typically denoted as "x". This rule states that if you have a function of the form \(f(x) = x^n\), where \(n\) is any real number, the derivative \(f'(x)\) will be \(nx^{n-1}\). This means:
As seen in our example, for \(7x^{-6}\), the derivative is found by multiplying \(-6\) with \(7\), resulting in \(-42\), and then decreasing the exponent by one to get \(-7\). This results in \(-42x^{-7}\).
- Multiply the exponent \(n\) by the coefficient of the term.
- Reduce the exponent \(n\) by one.
- The new term is your derivative.
As seen in our example, for \(7x^{-6}\), the derivative is found by multiplying \(-6\) with \(7\), resulting in \(-42\), and then decreasing the exponent by one to get \(-7\). This results in \(-42x^{-7}\).
Derivative
A derivative represents how a function's output changes as its input changes. It's essentially the slope of the function at any given point. In simpler terms, it tells us how fast or slow something is changing.
- It's a fundamental concept in calculus.
- It allows us to understand rates of change and to find solutions to various real-world problems.
- The process of finding a derivative is known as differentiation.
Exponents
Exponents are a way of expressing repeated multiplication of a number by itself. An exponent indicates how many times a base is used as a factor. For example, \(x^3\) means \(x \times x \times x\). Important points about exponents include:
Changing roots and negative exponents can make applying the power rule more straightforward.
- An exponent of zero (\(x^0\)) equals one.
- A negative exponent represents division or the reciprocal of the base raised to the absolute value of the exponent.
- Fractional exponents represent roots; for instance, \(x^{1/2}\) is the square root of \(x\).
Changing roots and negative exponents can make applying the power rule more straightforward.
Function Rewriting
Rewriting functions is a key step that simplifies differentiation. This involves expressing functions in a suitable form, like using exponents instead of roots. This transformation helps in applying differentiation rules efficiently.
Rewriting functions to a more manageable form is a crucial step before tackling differentiation.
- Transforms can involve changing square roots to fractional powers.
- Helps in breaking down complex expressions.
- Makes the application of the power rule easier.
Rewriting functions to a more manageable form is a crucial step before tackling differentiation.
