Summarize this article:
108 LearnersLast updated on September 27, 2025
Derivative of sin 5x
The derivative of sin(5x) helps us understand how the sine function changes in response to a slight change in x. Derivatives are useful in various real-life applications, including calculating profit or loss. We will now explore the derivative of sin(5x) in detail.
What is the Derivative of sin 5x?
The derivative of sin 5x is denoted as d/dx (sin 5x) or (sin 5x)', and its value is 5cos(5x). This indicates that the function sin 5x is differentiable within its domain.
Key concepts include:
Sine Function: sin(5x) represents the sine function with an argument of 5x.
Chain Rule: Used for differentiating composite functions like sin(5x).
Cosine Function: cos(x) is the derivative of sin(x), and it appears in the derivative formula for sin(5x).
Derivative of sin 5x Formula
The derivative of sin 5x can be represented as d/dx (sin 5x) or (sin 5x)'. Using the chain rule, the formula is: d/dx (sin 5x) = 5 cos(5x)
This formula applies for all x, given that sin 5x is a continuous function.
Proofs of the Derivative of sin 5x
We can derive the derivative of sin 5x using different methods.
These include:
- By First Principle
- Using the Chain Rule
By First Principle
To prove by the first principle, consider f(x) = sin 5x. Its derivative is expressed as the limit of the difference quotient: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Substituting f(x) = sin 5x, we have: f'(x) = limₕ→₀ [sin(5(x + h)) - sin(5x)] / h Applying the sine difference identity and limit laws, the result is: f'(x) = 5 cos(5x)
Using the Chain Rule
Let u = 5x, so that f(x) = sin(u). Then, f'(x) = cos(u) * du/dx. Since du/dx = 5, we have: d/dx (sin 5x) = cos(5x) * 5 = 5 cos(5x)
Higher-Order Derivatives of sin 5x
Higher-order derivatives are obtained by repeatedly differentiating a function. For sin 5x, the first derivative is 5 cos(5x), the second derivative is -25 sin(5x), and so on.
These derivatives help us understand changes at multiple levels, much like how acceleration (second derivative) relates to speed (first derivative).
Special Cases:
When x is 0, the derivative of sin 5x = 5 cos(0) = 5.
At multiples of π/5, the derivative equals 0 because cos(nπ) = 0 for n = ±1, ±2, ...
Common Mistakes and How to Avoid Them in Derivatives of sin 5x
When differentiating sin 5x, students often make errors. Understanding the correct approach can help. Here are common mistakes and solutions:
Mistake 1
Not applying the chain rule correctly
Students may forget the chain rule requires multiplying by the derivative of the inner function. For sin(5x), the inner derivative is 5.
Correct differentiation: d/dx (sin 5x) = 5 cos(5x).
Mistake 2
Ignoring the factor of 5
Some students forget to multiply by 5, leading to an incomplete result.
Always remember to apply the chain rule, which introduces this factor.
Mistake 3
Confusing sine and cosine derivatives
It's common to mix up the derivatives of sine and cosine.
Remember: the derivative of sin(x) is cos(x), and that of cos(x) is -sin(x).
Mistake 4
Overlooking special cases
Students might not recognize when the derivative simplifies to zero, such as at specific points like π/5.
Be mindful of the function's behavior at these points.
Mistake 5
Incorrectly using trigonometric identities
Errors occur when students misapply identities.
For example, incorrectly using sin2(x) + cos2(x) = 1 in differentiation contexts. Ensure proper usage of identities.
Examples Using the Derivative of sin 5x
Problem 1
Calculate the derivative of sin 5x·cos 5x
Let f(x) = sin 5x · cos 5x. Using the product rule, f'(x) = u'v + uv'. Here, u = sin 5x and v = cos 5x. Differentiating each, u' = 5 cos 5x and v' = -5 sin 5x. Substituting, we get f'(x) = (5 cos 5x)(cos 5x) + (sin 5x)(-5 sin 5x) = 5 cos²5x - 5 sin²5x.
Explanation
We applied the product rule by differentiating each part and combining the results.
Simplifying gives the final answer.
Problem 2
A pendulum swings according to the function y = sin(5x), where y is the displacement at time x. Find the rate of change of displacement when x = π/10.
The function is y = sin(5x). Differentiate to get dy/dx = 5 cos(5x). At x = π/10, dy/dx = 5 cos(5(π/10)) = 5 cos(π/2) = 0. The rate of change of displacement is 0 at x = π/10, indicating a momentary pause in motion.
Explanation
At x = π/10, the pendulum reaches a peak or trough, so the rate of change is zero, reflecting no motion at that instant.
Problem 3
Find the second derivative of y = sin 5x.
First, find dy/dx = 5 cos 5x. Then, the second derivative is d²y/dx² = d/dx(5 cos 5x) = -25 sin 5x.
Explanation
We first differentiated to find the first derivative, then differentiated again to find the second derivative, applying chain rules where necessary.
Problem 4
Prove: d/dx (sin²(5x)) = 10 sin(5x) cos(5x).
Let y = sin²(5x). Using the chain rule, dy/dx = 2 sin(5x) d/dx(sin(5x)) = 2 sin(5x) (5 cos(5x)) = 10 sin(5x) cos(5x).
Explanation
The chain rule was applied first to the outer function, then to the inner function, and results were multiplied together.
Problem 5
Differentiate (sin 5x)/x.
Using the quotient rule: d/dx [(sin 5x)/x] = [x(5 cos 5x) - sin 5x(1)] / x² = (5x cos 5x - sin 5x) / x².
Explanation
We used the quotient rule by identifying the numerator and denominator functions, differentiating each, and simplifying the result.
FAQs on the Derivative of sin 5x
1.What is the derivative of sin 5x?
2.How is the derivative of sin 5x applied in real life?
3.Is the derivative of sin 5x defined for all x?
4.How can the chain rule help with differentiating sin 5x?
5.What is the second derivative of sin 5x?
Important Glossaries for the Derivative of sin 5x
- Derivative: Describes how a function changes as its input changes.
- Sine Function: Denoted as sin(x), a basic trigonometric function.
- Cosine Function: Denoted as cos(x), derivative of sin(x).
- Chain Rule: A fundamental rule for differentiating composite functions.
- Quotient Rule: Used for differentiating ratios of functions.
Explore More calculus
![Important Math Links Icon]()
Previous to Derivative of sin 5x
![Important Math Links Icon]()
Next to Derivative of sin 5x
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
