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Differentiate e^{x^2 + sin x} with respect to x.

Question:

Differentiate {eq}e^{x^2 + \sin x} {/eq} with respect to {eq}x {/eq}.

Chain rule of differentiation:


Let us consider the composite function {eq}f(x) = g(h(x)) {/eq}, then the derivative of that composite function is given by,

{eq}f'(x) = g'(h(x)) \cdot h'(x) {/eq}


The power rule of differentiation states that the derivative of {eq}x^n {/eq} is {eq}nx^{n - 1} {/eq}, where {eq}n {/eq} is a real number.

Answer and Explanation:


Let {eq}f(x) = e^{x^2 + \sin x} {/eq},


Differentiating the function with respect to {eq}x {/eq},

$$\displaystyle \begin{align*} f'(x) &= \dfrac {d}{dx} \left [ e^{x^2 + \sin x} \right ] \\ &= e^{x^2 + \sin x} \cdot \dfrac {d}{dx} \left [ x^2 + \sin x \right ] \\ &= e^{x^2 + \sin x} \left ( 2x + \cos x \right ) \\ f'(x) &= \left ( 2x + \cos x \right ) e^{x^2 + \sin x} \\ \end{align*} $$


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Basic Calculus: Rules & Formulas

from Calculus: Tutoring Solution

Chapter 3 / Lesson 6
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