Question

Use the General Power Rule to find the derivative of the function. $$h(t)={(1-t^2)}^4$$

Short answer

The derivative of the function $h(t)=(1-t^2{)}^4$ is $h^{\prime }(t)=-8t(1-t^2{)}^3$.

Step-by-step solution

Identify the inner function

Identify the inner function $u(t)$ such that we can apply the chain rule. In this case, $u(t)=1-t^2$.

Differentiate the inner function

To apply the chain rule, we need to know the derivative of the inner function, $u^{\prime }(t)$. The derivative of $u(t)=1-t^2$ is $u^{\prime }(t)=0-2t=-2t$.

Apply the General Power Rule

The General Power Rule states that if $h(t)=u^n$, then $h^{\prime }(t)=n{u}^{n-1}{u}^{\prime }$, where $u^{\prime }=du/dt$ and $n$ is a real number. Here, $u(t)=1-t^2$ and $n=4$. Hence, applying the General Power Rule, $h^{\prime }(t)=4(1-t^2{)}^{4-1}\ast (-2t)$.

Simplify the result

Simplify the resulting derivative. So, our final derivative $h^{\prime }(t)=-8t(1-t^2{)}^3$.

Key concepts

General Power Rule

The General Power Rule is a helpful method for finding derivatives of functions that are raised to a power. In mathematical terms, if you have a function of the form $h(t)=u^n$, the derivative $h^{\prime }(t)$ can be calculated using the formula $n{u}^{n-1}{u}^{\prime }$. Here, $n$ represents the power the function is raised to, while $u^{\prime }$ represents the derivative of the inner function $u(t)$. This rule simplifies the differentiation process when combined with the chain rule. It is especially useful for expressions where the entire function is raised to a power, as is the case in our exercise.
To see this in action, consider the function $h(t)=(1-t^2{)}^4$. Applying the General Power Rule, we calculate $4(1-t^2{)}^3(-2t)$, resulting in $-8t(1-t^2{)}^3$ after simplification. This exemplifies its practical usage.

chain rule

The chain rule is a fundamental method in calculus used to differentiate composite functions. This rule is crucial when dealing with functions that nest other functions within them, which we call inner functions.
When a function is composed of another, such as $h(t)=(1-t^2{)}^4$, the chain rule helps find its derivative. It essentially states that to find the derivative of a composite function, you multiply the derivative of the outer function by the derivative of the inner function.
In this example, you first differentiate the outer function $(1-t^2{)}^4$ using the General Power Rule, resulting in $4(1-t^2{)}^3$. Then, you multiply it by the derivative of the inner function, $-2t$. Combining these results gives the final derivative: $-8t(1-t^2{)}^3$. The chain rule is indispensable, allowing you to neatly unravel complex nested functions.

differentiation

Differentiation is the mathematical process of finding the derivative of a function. Derivatives represent the rate of change of a function with respect to its variable. This foundational concept of calculus plays a crucial role in various fields such as physics, engineering, and economics.
In the context of our exercise, differentiation of the function $(1-t^2{)}^4$ involves a few steps. First, recognizing the inner function and applying the General Power Rule. However, to apply these rules effectively, one must be comfortable with the basic differentiation techniques and formulas.
For simpler tasks, calculating derivatives might involve straightforward rules like the product, quotient, or power rules. But with composite functions, like in our example, we employ the chain rule alongside these rules. Understanding and applying differentiation correctly leads to solutions that can predict behavior and motion regarding changes over time or other variables.

inner function

The term 'inner function' refers to a function nestled within another function in composite expressions. Identifying the inner function is crucial when applying the chain rule during differentiation.
In our exercise, the function $h(t)=(1-t^2{)}^4$ consists of an outer function $u^4$ and an inner function $u(t)=1-t^2$. Successfully identifying and differentiating this inner function is vital to solving the derivative of the entire expression.
By finding its derivative $u^{\prime }(t)=-2t$, you prepare yourself to apply the General Power Rule correctly. Recognizing the correct inner and outer functions simplifies the differentiation process, making it more manageable and, eventually, second nature. Understanding this concept allows for greater accuracy and confidence when working with more complex functions.