Derivative of a Power Function

Find the derivative of $(5x + 3)^4$.

When you encounter a problem asking for the derivative of an expression raised to a power, such as $(5x + 3)^4$, it is crucial to apply the chain rule, a fundamental tool in calculus. The chain rule allows you to differentiate composite functions by differentiating the outer function and then multiplying it by the derivative of the inner function. Essentially, you take the derivative of the outside, keeping the inside unchanged, and then multiply by the derivative of the inside function. This structured approach is vital in handling complex expressions like this one.

Additionally, understanding the product and power rules can assist in breaking down such problems. The power rule mandates that you bring down the exponent as a coefficient and then decrease the exponent by one. In combination with the chain rule, it enables the differentiation of more complicated forms, often seen in algebra and calculus curricula.

Problems of this nature are common in both single-variable calculus and its applications in real-world situations where rates of change are examined. Mastery of these techniques not only leads to solving derivative problems efficiently but also enhances the understanding of how functions behave, which is fundamental in advanced mathematics and engineering fields.

Related Problems