Problem 32
Question
Use the definition of the derivative to show that \(D_{x}(\sin 5 x)=5 \cos 5 x .\)
Step-by-Step Solution
Verified Answer
The derivative of \(\sin(5x)\) is \(5\cos(5x)\).
1Step 1: Recall the Definition of the Derivative
The derivative of a function \(f(x)\) at point \(x\) is given by the limit \(\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\). We will apply this for \(f(x) = \sin(5x)\).
2Step 2: Setup the Expression for \(f(x+h)\)
Replacing \(x\) with \(x + h\) in the function \(f(x) = \sin(5x)\), we get \(f(x+h) = \sin(5(x+h)) = \sin(5x + 5h)\).
3Step 3: Apply the Definition of the Derivative
Substitute \(f(x)\) and \(f(x+h)\) in the derivative formula to get the expression: \[D_x(\sin(5x)) = \lim_{h \to 0} \frac{\sin(5x + 5h) - \sin(5x)}{h}.\]
4Step 4: Use the Sine Difference Identity
The identity \(\sin(a) - \sin(b) = 2 \cos\left(\frac{a+b}{2}\right) \sin\left(\frac{a-b}{2}\right)\) can simplify our expression. For \(a = 5x + 5h\) and \(b = 5x\), we get: \[\sin(5x + 5h) - \sin(5x) = 2 \cos\left(5x + \frac{5h}{2}\right) \sin\left(\frac{5h}{2}\right).\]
5Step 5: Simplify the Limit Expression
Substitute the identity into the limit expression:\[\lim_{h \to 0} \frac{2 \cos\left(5x + \frac{5h}{2}\right) \sin\left(\frac{5h}{2}\right)}{h}.\]Using the fact that \(\sin\left(\frac{5h}{2}\right) \approx \frac{5h}{2}\) as \(h \to 0\), the expression simplifies further to:\[\lim_{h \to 0} \cos(5x + \frac{5h}{2}) \cdot 5 \left(\frac{\sin(\frac{5h}{2})}{\frac{5h}{2}}\right) = 5\cos(5x).\]
6Step 6: Use the Limit Property of \(\frac{\sin(x)}{x}\)
The known property \(\lim_{u \to 0} \frac{\sin(u)}{u} = 1\) ensures that:\[\lim_{h \to 0} \cos(5x + \frac{5h}{2}) \cdot 5 \cdot 1 = 5 \cos(5x).\] Here, the term \(\cos(5x + \frac{5h}{2})\) approaches \(\cos(5x)\) as \(h\) approaches 0.
Key Concepts
Trigonometric FunctionsLimit Definition of DerivativeSine and Cosine Identities
Trigonometric Functions
Trigonometric functions play a critical role in understanding phenomena that involve angles and periodic patterns. Two of the primary trigonometric functions are the sine function, denoted as \( \sin(x) \), and the cosine function, denoted as \( \cos(x) \). These functions are periodic and relate the angles and sides of right triangles.
Understanding these functions helps in many fields such as physics, engineering, and even daily life, for calculating elements like sound and light waves.
- Sine Function (\( \sin(x) \)): Represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
- Cosine Function (\( \cos(x) \)): Represents the ratio of the length of the adjacent side to the hypotenuse in a right-angle triangle.
Understanding these functions helps in many fields such as physics, engineering, and even daily life, for calculating elements like sound and light waves.
Limit Definition of Derivative
The derivative of a function, at its essence, measures how the function changes as its input changes. The limit definition of the derivative provides a foundational approach to calculating this rate of change.
The formal definition of the derivative of a function \( f(x) \) at a point \( x \) is given by the formula:\[\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]
This expression calculates the slope of the tangent line to the curve of \( f(x) \) at the point \( x \). By examining how the function value \( f(x+h) \) converges to \( f(x) \) as \( h \) approaches zero, we can determine how steeply \( f(x) \) is increasing or decreasing at \( x \).
The limit definition is crucial for gaining deep insights into the behavior of functions, especially when they involve more complex operations like trigonometric or exponential functions.
The formal definition of the derivative of a function \( f(x) \) at a point \( x \) is given by the formula:\[\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]
This expression calculates the slope of the tangent line to the curve of \( f(x) \) at the point \( x \). By examining how the function value \( f(x+h) \) converges to \( f(x) \) as \( h \) approaches zero, we can determine how steeply \( f(x) \) is increasing or decreasing at \( x \).
The limit definition is crucial for gaining deep insights into the behavior of functions, especially when they involve more complex operations like trigonometric or exponential functions.
Sine and Cosine Identities
Sine and cosine identities are amazingly useful tools that simplify complex expressions involving trigonometric functions. These identities allow us to express trigonometric functions in terms of each other and simplify expressions significantly.
One essential identity used in calculus problems involving derivatives, such as the derivative of \( \sin(5x) \), is the sine difference identity:\[\sin(a) - \sin(b) = 2 \cos\left(\frac{a + b}{2}\right) \sin\left(\frac{a - b}{2}\right)\]
This particular identity transforms a difference between two sine terms into a product involving cosine and sine, facilitating simplification in calculus problems. For example, when finding the derivative of \( \sin(5x) \) using this identity, it helps us to rewrite and simplify the limit expression so that it can be easily evaluated.
By understanding these identities, we can tackle more intricate calculus problems and develop a deeper comprehension of the behavior of periodic functions.
One essential identity used in calculus problems involving derivatives, such as the derivative of \( \sin(5x) \), is the sine difference identity:\[\sin(a) - \sin(b) = 2 \cos\left(\frac{a + b}{2}\right) \sin\left(\frac{a - b}{2}\right)\]
This particular identity transforms a difference between two sine terms into a product involving cosine and sine, facilitating simplification in calculus problems. For example, when finding the derivative of \( \sin(5x) \) using this identity, it helps us to rewrite and simplify the limit expression so that it can be easily evaluated.
By understanding these identities, we can tackle more intricate calculus problems and develop a deeper comprehension of the behavior of periodic functions.
Other exercises in this chapter
Problem 31
Draw the graph of \(y=f(x)=x^{3}-2 x^{2}+1\). Then find the slope of the tangent line at (a) \(-1\) (b) 0 (c) 1 (d) \(3.2\)
View solution Problem 31
In Problems 29-32, evaluate the indicated derivative. $$ F^{\prime}(1) \text { if } F(t)=\sin \left(t^{2}+3 t+1\right) $$
View solution Problem 32
The positions of two objects, \(P_{1}\) and \(P_{2}\), on a coordinate line at the end of \(t\) seconds are given by \(s_{1}=3 t^{3}-12 t^{2}+\) \(18 t+5\) and
View solution Problem 32
Find the indicated derivative. \(y^{\prime}\) if \(y=e^{x / \ln x}\)
View solution