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Derivative of $$$x^{3} + y^{5}$$$ with respect to $$$y$$$

The calculator will find the derivative of $$$x^{3} + y^{5}$$$ with respect to $$$y$$$, with steps shown.

Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps

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Your Input

Find $$$\frac{d}{dy} \left(x^{3} + y^{5}\right)$$$.

Solution

The derivative of a sum/difference is the sum/difference of derivatives:

$${\color{red}\left(\frac{d}{dy} \left(x^{3} + y^{5}\right)\right)} = {\color{red}\left(\frac{d}{dy} \left(x^{3}\right) + \frac{d}{dy} \left(y^{5}\right)\right)}$$

The derivative of a constant is $$$0$$$:

$${\color{red}\left(\frac{d}{dy} \left(x^{3}\right)\right)} + \frac{d}{dy} \left(y^{5}\right) = {\color{red}\left(0\right)} + \frac{d}{dy} \left(y^{5}\right)$$

Apply the power rule $$$\frac{d}{dy} \left(y^{n}\right) = n y^{n - 1}$$$ with $$$n = 5$$$:

$${\color{red}\left(\frac{d}{dy} \left(y^{5}\right)\right)} = {\color{red}\left(5 y^{4}\right)}$$

Thus, $$$\frac{d}{dy} \left(x^{3} + y^{5}\right) = 5 y^{4}$$$.

Answer

$$$\frac{d}{dy} \left(x^{3} + y^{5}\right) = 5 y^{4}$$$A

Related Notes: Definition of Derivative , Derivatives of Elementary Functions , Table of Derivatives , Related Rates , Studying Derivative Graphically , Optimization Problems , Applications to Economics , Constant Multiple Rule , Sum and Difference Rules , Product Rule , Quotient Rule , Chain Rule , Derivative of Inverse Function