Question

Find the derivative of function $y = {x^2}{e^x}\sin x$ .
a) \[{e^x}x\left[ {2\sin x + x\sin x + x\cos x} \right]\]
b) \[{e^x}x\left[ {2\sin x + x\sin x - \cos x} \right]\]
c) \[{e^x}x\left[ {2\sin x + x\sin x + \cos x} \right]\]
d) None of these

Answer 1

Hint: In the given problem, we are required to differentiate $y = {x^2}{e^x}\sin x$ with respect to x. Since, $y = {x^2}{e^x}\sin x$ is a product function, so we will have to apply product rule of differentiation in the process of differentiating $y = {x^2}{e^x}\sin x$ . Also derivatives of basic algebraic and trigonometric functions must be remembered thoroughly.


Complete step by step solution: 
To find derivative of $y = {x^2}{e^x}\sin x$ with respect to $x$ , we have to find differentiate $y = {x^2}{e^x}\sin x$ with respect to $x$.
So, Derivative of $y = {x^2}{e^x}\sin x$ with respect to $x$ can be calculated as $\dfrac{d}{{dx}}\left( {{x^2}{e^x}\sin x} \right)$ .
Now, \[\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{x^2}{e^x}\sin x} \right)\] .
Now, using the product rule of differentiation, we know that $\dfrac{d}{{dx}}\left( {f(x) \times g(x)} \right) = f(x) \times \dfrac{d}{{dx}}\left( {g(x)} \right) + g(x) \times \dfrac{d}{{dx}}\left( {f(x)} \right)$ .
So, applying product rule to $\dfrac{d}{{dx}}\left( {{x^2}{e^x}\sin x} \right)$, we get,
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^x}\sin x\dfrac{d}{{dx}}\left( {{x^2}} \right) + {x^2}\dfrac{d}{{dx}}\left( {{e^x}\sin x} \right)\]
Using power rule of differentiation, we get,
\[ \Rightarrow \dfrac{{dy}}{{dx}} = 2x{e^x}\sin x + {x^2}\dfrac{d}{{dx}}\left( {{e^x}\sin x} \right)\]
Now, we again apply the product rule of differentiation. So, we get,
\[ \Rightarrow \dfrac{{dy}}{{dx}} = 2x{e^x}\sin x + {x^2}\left[ {\sin x\dfrac{d}{{dx}}\left( {{e^x}} \right) + {e^x}\dfrac{d}{{dx}}\left( {\sin x} \right)} \right]\]
Now, we know that the derivative of sine is cosine and the derivative of ${e^x}$ is ${e^x}$.
So, we get,
\[ \Rightarrow \dfrac{{dy}}{{dx}} = 2x{e^x}\sin x + {x^2}\left[ {\sin x \times \left( {{e^x}} \right) + {e^x} \times \left( {\cos x} \right)} \right]\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^x}x\left[ {2\sin x + x\sin x + x\cos x} \right]\]
Hence, the correct answer is option (a).

Note: The derivatives of basic trigonometric and algebraic functions must be learned by heart in order to find derivatives of complex composite functions using product rule and chain rule of differentiation. The product rule of differentiation involves differentiating a product of two functions and the chain rule of differentiation involves differentiating a composite by introducing new unknowns to ease the process and examine the behavior of function layer by layer.