derivative of tan(5x)
Understand the Problem
The question is asking to find the derivative of the function tan(5x). This involves applying the derivative rules in calculus, specifically the chain rule, since 5x is a composite function.
Answer
$$ \frac{d}{dx}(\tan(5x)) = 5 \sec^2(5x) $$
Answer for screen readers
$$ \frac{d}{dx}(\tan(5x)) = 5 \sec^2(5x) $$
Steps to Solve
- Identify the function and its components
Our function is $f(x) = \tan(5x)$. We can see that this is a composition of two functions: the tangent function and the inner function $g(x) = 5x$.
- Apply the derivative of the outer function
The derivative of $\tan(u)$ with respect to $u$ is $\sec^2(u)$. In our case, $u = 5x$, so we write:
$$ \frac{d}{dx}(\tan(5x)) = \sec^2(5x) $$
- Apply the chain rule
We need to apply the chain rule, which states that we should multiply the derivative of the outer function by the derivative of the inner function. The derivative of $g(x) = 5x$ is simply $5$:
$$ \frac{d}{dx}(5x) = 5 $$
- Combine the results
Now, we can combine both results using the chain rule:
$$ \frac{d}{dx}(\tan(5x)) = \sec^2(5x) \cdot 5 $$
So the complete derivative is:
$$ \frac{d}{dx}(\tan(5x)) = 5 \sec^2(5x) $$
$$ \frac{d}{dx}(\tan(5x)) = 5 \sec^2(5x) $$
More Information
The derivative $5 \sec^2(5x)$ shows how the rate of change of the function $\tan(5x)$ responds to changes in $x$. The $\sec^2(5x)$ part indicates how steep the tangent function is at that point, while the factor of 5 represents the impact of the inner function's scaling.
Tips
- Forgetting to apply the chain rule: It's common to take the derivative of the outer function only without considering the inner function's derivative.
- Miscalculating the derivative of $5x$: It's important to remember that the derivative of a linear function like $5x$ is simply its coefficient, which is 5.
